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Foundations of Schrödinger Bridges for Generative Modeling
Sophia Tang
Intelligence
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Summary
This paper provides a comprehensive mathematical foundation for Schrödinger bridges (SB) as a unifying framework for modern generative modeling, including diffusion models, score-based models, and flow matching. It connects optimal transport, stochastic control, and path-space optimization to define optimal stochastic bridges between probability distributions, offering a toolkit for constructing these bridges and their computational implementations.
Entities (6)
Relation Signals (4)
Schrödinger Bridge → isbasedon → Optimal Transport
confidence 100% · drawing on optimal transport, stochastic control, and path-space optimization
Stochastic Optimal Control → reformulates → Schrödinger Bridge
confidence 100% · reformulates the SB problem through the lens of stochastic optimal control (SOC) theory
Sinkhorn’s Algorithm → solves → Static Schrödinger Bridge Problem
confidence 100% · breaks down the classic algorithm for solving the static SB problem, known as Sinkhorn’s algorithm
Schrödinger Bridge → unifies → Generative Modeling
confidence 100% · Schrödinger bridges provide a unifying principle underlying these approaches [generative modeling]
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Abstract
Abstract:At the core of modern generative modeling frameworks, including diffusion models, score-based models, and flow matching, is the task of transforming a simple prior distribution into a complex target distribution through stochastic paths in probability space. Schrödinger bridges provide a unifying principle underlying these approaches, framing the problem as determining an optimal stochastic bridge between marginal distribution constraints with minimal-entropy deviations from a pre-defined reference process. This guide develops the mathematical foundations of the Schrödinger bridge problem, drawing on optimal transport, stochastic control, and path-space optimization, and focuses on its dynamic formulation with direct connections to modern generative modeling. We build a comprehensive toolkit for constructing Schrödinger bridges from first principles, and show how these constructions give rise to generalized and task-specific computational methods.
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- Source: https://arxiv.org/abs/2603.18992v1
- Canonical: https://arxiv.org/abs/2603.18992v1
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Foundations of Schr ̈odinger Bridges for Generative Modeling Sophia Tang Department of Computer and Information Science University of Pennsylvania Correspondence to: sophtang@seas.upenn.edu Abstract At the core of modern generative modeling frameworks—including diffusion models, score-based models, and flow matching—is the task of transforming a simple prior distribution into a complex target distribution through stochastic paths in probability space. Schr ̈odinger bridges provide a unifying principle underlying these approaches, framing the problem as determining an optimal stochastic bridge between marginal distribution constraints with minimal-entropy deviations from a pre-defined reference process. This guide develops the mathematical foundations of the Schr ̈odinger bridge problem, drawing on optimal transport, stochastic control, and path-space optimization, and focuses on its dynamic formulation with direct connections to modern generative modeling. We build a comprehensive toolkit for constructing Schr ̈odinger bridges from first principles, and show how these constructions give rise to generalized and task-specific computational methods. 1 arXiv:2603.18992v1 [cs.LG] 19 Mar 2026 Contents 1 The Static Schr ̈odinger Bridge Problem11 1.1 The Optimal Mass Transport Problem . . . . . . . . . . . . . . . . . . . . . . . . . .11 1.2 Entropy on Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 1.3 Entropic Optimal Transport Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .14 1.4 Static Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 1.5 Sinkhorn’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 1.6 Closing Remarks for Section 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 2 The Dynamic Schr ̈odinger Bridge Problem28 2.1 Dynamic Optimal Transport Problem . . . . . . . . . . . . . . . . . . . . . . . . . .28 2.2 Dynamic Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.3 Path Measures and Itˆo Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 2.4 Fokker-Planck and Feynman-Kac Equations . . . . . . . . . . . . . . . . . . . . . . .35 2.5 Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 2.6 Path Measure Radon-Nikodym Derivative and KL Divergence . . . . . . . . . . . . .44 2.7 Schr ̈odinger Bridge with Arbitrary Reference Dynamics . . . . . . . . . . . . . . . .47 2.8 Hopf-Cole Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52 2.9 Schr ̈odinger Bridges as Entropy-Regularized Dynamic Optimal Transport . . . . . .57 2.10 Closing Remarks for Section 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 3 Schr ̈odinger Bridge Problem as Optimal Control62 3.1 Stochastic Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 3.2 Schr ̈odinger Bridges with Stochastic Optimal Control . . . . . . . . . . . . . . . . . .72 3.3 Objectives for Solving the SOC Problem . . . . . . . . . . . . . . . . . . . . . . . . .75 3.4 Closing Remarks for Section 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 4 Building Schr ̈odinger Bridges82 4.1 Mixture of Conditional Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 4.2 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86 4.3 Forward-Backward Stochastic Differential Equations . . . . . . . . . . . . . . . . . .91 4.4 Doob’s h-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 4.5 Markovian and Reciprocal Projections . . . . . . . . . . . . . . . . . . . . . . . . . .99 4.6 Stochastic Interpolants to Schr ̈odinger Bridges . . . . . . . . . . . . . . . . . . . . . 110 4.7 Closing Remarks for Section 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5 Variations of the Schr ̈odinger Bridge Problem118 5.1 Gaussian Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Generalized Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3 Multi-Marginal Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . 134 5.4 Unbalanced Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.5 Branched Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.6 Fractional Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.7 Closing Remarks for Section 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6 Generative Modeling with Schr ̈odinger Bridges152 6.1 A Primer on Score-Based Generative Modeling . . . . . . . . . . . . . . . . . . . . . 152 6.2 Likelihood Training of Forward-Backward SDEs . . . . . . . . . . . . . . . . . . . . . 153 6.3 Diffusion Schr ̈odinger Bridge Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.4 Simulation-Free Score and Flow Matching . . . . . . . . . . . . . . . . . . . . . . . . 160 6.5 Schr ̈odinger Bridge with Adjoint Matching . . . . . . . . . . . . . . . . . . . . . . . . 163 2 6.6 Closing Remarks for Section 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7 From Continuous to Discrete State Space180 7.1 Continuous-Time Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.2 Discrete Schr ̈odinger Bridge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.3 Stochastic Optimal Control of CTMCs . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.4 Discrete Schr ̈odinger Bridges with Stochastic Optimal Control . . . . . . . . . . . . . 194 7.5 Discrete Markov and Reciprocal Projections . . . . . . . . . . . . . . . . . . . . . . . 197 7.6 Discrete Diffusion Schr ̈odinger Bridge Matching . . . . . . . . . . . . . . . . . . . . . 204 7.7 Closing Remarks for Section 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8 Applications of Generative Modeling with Schr ̈odinger Bridges208 8.1 Data Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.2 Modeling Single-Cell State Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.3 Sampling Boltzmann Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.4 Closing Remarks for Section 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9 Conclusion215 3 Preface The field of generative modeling has experienced rapid and transformative progress in recent years. Advances range from foundational theoretical developments—such as diffusion models and flow matching—to algorithmic improvements in sampling speed and generation quality, as well as significant applications spanning language, video, and scientific domains. While this breadth of progress has driven remarkable innovation, it has also made the field increasingly complex and difficult to navigate, particularly from a foundational perspective. In this guide, we introduce Schr ̈odinger bridges as a unifying theoretical framework for generative modeling. This perspective generalizes a broad class of modern approaches—including diffusion models, flow-matching methods, and stochastic control formulations—while providing a principled and flexible foundation for addressing specialized scientific problems. The body of work that has built the Schr ̈odinger bridge framework spans multiple fields, from theoretical developments that formalize path-space optimization and entropy-regularized transport, to principled extensions in diverse problem settings, to algorithmic advances that enable scalable training and inference. Despite this progress, the literature remains fragmented, and the relationships between different formulations are often difficult to identify. The goal of this guide is to build both intuition and a deep mathematical understanding of the core principles of Schr ̈odinger bridges, from its origins in optimal transport to its dynamic path space formulation, which underlies modern generative modeling frameworks. At a high level, we begin from a single unifying principle: optimal stochastic bridges between distributions can be characterized as minimal-entropy deviations from a reference process subject to marginal constraints. From this perspective, we develop the mathematical tools spanning optimal transport, probability theory, and stochastic calculus required to understand the static and dynamic formulations of the Schr ̈odinger bridge problem, before naturally extending these ideas to diverse problem settings and modern generative modeling techniques. Throughout this guide, we construct a principled toolkit for building Schr ̈odinger bridges from first principles and show how these constructions give rise to theoretically-grounded and scalable computational methods for simulating the optimal Schr ̈odinger bridge dynamics. This guide contains eight primary sections, which are organized as follows: Section 1 introduces the static Schr ̈odinger bridge (SB) problem, which provides the theoretical foundations for optimal transport between probability distributions. •Section 1.1 traces back to the classical optimal mass transport (OMT) problem posed by Monge and Kantorovich, which seeks an optimal static coupling between probability distributions that minimizes total transport cost. • Section 1.2 introduces the foundational properties of entropy on probability spaces, including the fundamental properties of the Kullback-Leibler (KL) divergence. •Section 1.3 introduces the entropic optimal transport (EOT) problem, which leverages entropy as a method of regularizing the OMT problem with a pre-defined reference coupling, which yields a unique solution. •Section 1.4 extends the EOT problem to formalize the static Schr ̈odinger bridge problem and its corresponding dual problem, which introduces a pair of Schr ̈odinger potentials that uniquely yield a clean form of the optimal solution. • Section 1.5 breaks down the classic algorithm for solving the static SB problem, known as Sinkhorn’s algorithm, which alternates between optimizing the dual Schr ̈odinger potentials. 4 Section 2 lifts the static SB problem into the space of stochastic path measures defined as the dynamic Schr ̈odinger bridge (SB) problem. •Section 2.1 starts by redefining the static SB problem as learning a continuous-time deterministic flow between distributions, known as the dynamic optimal transport (OT) problem. •Section 2.7 introduces the dynamic Schr ̈odinger bridge (SB) problem, which reformulates the SB problem as an entropy minimization over stochastic processes. • Section 2.3 provides the key definitions and theory required for understanding path measures as stochastic differential equations (SDEs) that can be steered via control drifts and transformed through functions. •Section 2.4 derives the Fokker-Planck equation, governing how the probability density over the stochastic paths generated from an SDE evolves forward in time, and the Feynman-Kac equation, governing how functions evaluated at the end of a stochastic process evolve backward in time. •Section 2.5 derives Girsanov’s theorem from first principles, which allows us to define changes in measure and KL divergences on path space. •Section 2.6 explicitly breaks down the theory of Radon-Nikodym derivatives between path measures, which is used to explicitly define the relative-entropy between path measures. Section 3 reformulates the SB problem through the lens of stochastic optimal control (SOC) theory. • Section 3.1 introduces the general framework of stochastic optimal control, including Bellman’s Principle of Optimality given a terminal constraint and deriving the value function that defines the optimal control. • Section 3.2 connects SB to SOC, showing that the optimal bridge corresponds to an optimal control drift. • Section 3.3 develops practical objectives for solving the SOC problem. Section 4 several complementary mechanisms for building Schr ̈odinger bridges. • Section 4.1 introduces mixtures of conditional bridges constructed with the reference process and a pre-defined endpoint coupling. •Section 4.2 derives the time-reversal formula of SDEs, which is fundamental to backward dynamics in SB. •Section 4.3 introduces forward-backward stochastic differential equations (FBSDEs), providing a coupled characterization of SB with respect to the time-dependent Schr ̈odigner potentials. • Section 4.4 presents Doob’sh-transform, which constructs conditioned stochastic processes by tilting the reference process using h-function. •Section 4.5 formalizes Markovian and reciprocal projections, which perform entropy- minimizing projections in path space that converge to the optimal SB measure. •Section 4.6 introduces stochastic interpolants, providing a practical way to construct bridges between distributions as deterministic interpolants with Gaussian noise. 5 Section 5 explores important variations of the Schr ̈odinger bridge problem for different modeling settings. • Section 5.1 studies the Gaussian SB problem, which admits a closed-form solutions. •Section 5.2 introduces the generalized SB problem which generalizes dynamic SB to model mean-field interactions. •Section 5.3 introduces the multi-marginal SB problem, which extends SB to settings with multiple intermediate marginal constraints. •Section 5.4 develops the unbalanced SB problem, allowing mass creation and destruction along the stochastic trajectories. •Section 5.5 introduces the branched SB problem, enabling modeling of diverging trajectories to multiple distinct terminal modes. •Section 5.6 studies fractional SB problems, incorporating long-range temporal dependencies through fractional Brownian motion. Section 6 connects SB theory to modern generative modeling frameworks. •Section 6.1 provides a primer on score-based generative modeling, which learns gradients of log-densities. •Section 6.2 introduces likelihood training of forward-backward SDEs, which trains the forward and backward potential drifts with likelihood objectives. • Section 6.3 develops diffusion Schr ̈odinger bridge matching, which parameterizes the Iterative Markovian Fitting procedure with a learned Markov drift. •Section 6.4 introduces simulation-free score and flow matching, enabling training of SB without simulating full trajectories. • Section 6.5 presents adjoint matching, which learns the optimal SB while avoiding explicit sampling from target distributions. Section 7 extends SB to discrete state spaces. • Section 7.1 introduces continuous-time Markov chains (CTMCs) as discrete analogues of stochastic processes. •Section 7.2 formulates the discrete Schr ̈odinger bridge problem, introducing Radon- Nikodym derivatives and KL divergences for CTMCs. •Section 7.3 develops stochastic optimal control for CTMCs as a method of learning the optimal CTMC path measure with terminal constraint. •Section 7.4 connects discrete SB with stochastic optimal control and introduces practical algorithms and objectives. • Section 7.5 introduces Markovian and reciprocal projections in discrete spaces. •Section 7.6 develops discrete diffusion Schr ̈odinger bridge matching, which parameterizes the Markovian and reciprocal generators to solve the discrete SB. 6 Section 8 highlights diverse applications of generative modeling with Schr ̈odinger bridges. •Section 8.1 applies SB to data translation tasks, which map between structured data distributions. •Section 8.2 studies single-cell state dynamics, showing how SB formulations can model cell population dynamics and responses to perturbation. •Section 8.3 applies SB to sampling Boltzmann distributions, showing how the SB frame- works can generate from unnormalized energy distributions without explicit samples. This guide is designed to establish both an intuitive and logical flow between foundational concepts, paired with rigorous derivations that solidify abstract ideas with principled theory. Despite the mathematically rigorous exploration of Schr ̈odinger bridges in this guide, it does not require prior background on stochastic calculus, Schr ̈odinger bridges, or generative modeling, as all necessary concepts are clearly defined and explained within the guide. Given this, the guide is designed for researchers, practitioners, and students interested in gaining a deep foundational understanding of generative modeling and its connections with recent developments in the field. This guide is not intended to be an exhaustive survey of all algorithms, architectures, or empirical applications involving Schr ̈odinger bridges. Instead, our goal is to provide readers with a conceptual and mathematical foundation from which diverse models and future developments can be understood. 7 Notation Throughout this guide, we often refer to the control drift and velocity field of an SDE. While it will become more apparent while reading, the control drift refers to added term to the reference drift which is scaled by the diffusion coefficient, generally denotedσ t u(x,t), and the velocity field refers to the entire non-diffusion term in the SDE that appears beforedtincluding both the reference drift and the control drift, generally denoted v(x,t). A key notational convention we adopt for clarity is using theX u 0:T = (X t ) u t∈[0,T ] when we want to emphasize that the path measure is generated under a specific control driftu(x,t), which appears in the sections on stochastic optimal control. In other sections, we omit this superscript when the underlying path measure is clear from context. In the table below, we list the common notation used throughout this guide, which we will often define repeatedly throughout the text for clarity. NotationMeaning Basic Notation KL(·∥·)Kullback-Leibler divergence infinfimum or the largest lower bound of a set supsupremum or the smallest upper bound of a set ∂ t partial derivative with respect to t d,dt,dxdifferential, time differential, state differential ∇gradient operator, shorthand for ∇ unless specified otherwise ∇·divergence operator, shorthand for ∇· unless specified otherwise ∆ =∇·∇ =∇ 2 Laplacian operator, shorthand for ∆ x · unless specified otherwise φ∈ C 2 (R d ) functions that are twice continuously differentiable with respect to x∈ R d φ∈ C 2,1 (R d × [0,T ])functions that are twice continuously differentiable with respect to x∈ R d and continuously differentiable with respect to t∈ [0,T ] φ∈ L 2 square integrable functions with finite L 2 norm I d d-dimensional Identity matrix 1 x=y indicator function that returns 1 when x = y and 0 otherwise δL δu functional derivative of the functional L with respect to u Matrix and Vector Operations ⟨·,·⟩inner sum ∥·∥ 2 L 2 -norm unless specified otherwise A ⊤ vector or matrix transpose A −1 matrix inverse State Spaces X,Yarbitrary state spaces (in Section 1,Xis the source state space and Yis the target state space and in Section 7,X:=1,...,dis used to denote a finite state space) X ×Yproduct space of X and Y R d d-dimensional state space P(·)probability space 8 P(R d )space of probability distributions over the state space R d C([0,T ]; R d )path space over time horizon t∈ [0,T ] and state space R d P(C([0,T ]; R d ))space of probability distributions over the path space Uspace of feasible control drifts, defined in Definition 2.22 Probability Densities p t marginal density of path measure at timet, typically the controlled path measure P u p 0 ,p T initial and terminal marginals generated from path measure P p ⋆ t marginal density of optimal path measure P ⋆ at time t p ⋆ 0 ,p ⋆ T initial and terminal marginals generated from optimal path measure P q t marginal density of reference path measure Q at time t q 0 ,q T initial and terminal marginals generated from reference path mea- sure Q ̃p s marginal density with respect to reverse time coordinates:=T − t π 0 marginal distribution constraint at time t = 0 π T marginal distribution constraint at time t = T π 0,T endpoint law or coupling distribution π 0 ⊗ π T product measure π ⋆ 0,T optimal OT or SB endpoint law or coupling distribution Stochastic Processes and Path Measures xrealized state in R d X t random variable in R d X 0:T , (X t ) t∈[0,T ] forward stochastic process over time t∈ [0,T ] in R d ̃ X 0:T , ( ̃ X s ) s∈[0,T ] backward stochastic process over reverse times=T − t∈[0,T] in R d X u 0:T , (X u t ) t∈[0,T ] forward stochastic process generated from the controlled SDE with control u B t Brownian motion random variable P, Qpath measure in P(C([0,T ]; R d )) P 0,T , Q 0,T , P ⋆ 0,T , P u 0,T , Π 0,T , M 0,T endpoint law of path measure P u controlled path measure in P(C([0,T ]; R d )) with control drift u Qreference path measure that defines the prior dynamics σBpure Brownian motion path measure with SDE dX t = σ t dB t P ⋆ path measure of the optimal solution Π mixture of bridges under the reference process defined as Π := Π 0,T Q ·|0,T R(Q)reciprocal class of Q containing all mixtures of bridges MMarkov path measure Mspace of Markov measures M ⋆ := proj M (·)Markov projection of a reciprocal measure Π ⋆ := proj R(Q) (·)reciprocal projection of a reciprocal measure 9 P ̃u ≪ P u absolute continuity, whereP ̃u is absolutely continuous with respect to P u P ̃u ∼ P u mutual absolute continuity, whereP ̃u is absolutely continuous with respect to P u , and vice versa P τ|t (x τ |x) or P(X τ = x τ |X t = x) transition density from state x at time t to state x τ at time τ Schr ̈odinger Bridge and Generative Modeling Theory f (x,t) : R d × [0,T ]→ R d uncontrolled drift of reference process Q φ :X → R, ˆφ :Y → Rstatic Schr ̈odinger potentials defined on X and Y φ⊕ ˆφseparable sum of functions defined on two coordinates φ t (x) : R d × [0,T ]→ Rforward Schr ̈odinger bridge potential ˆφ t (x) : R d × [0,T ]→ Rbackward Schr ̈odinger bridge potential u(x,t) : R d × [0,T ]→ R d control drift u ⋆ (x,t) : R d × [0,T ]→ R d Schr ̈odinger bridge drift or optimal control drift ̄ u(x,t) : R d × [0,T ]→ R d non-gradient-tracking control drift ̄ u = stopgrad(u) v(x,t) : R d × [0,T ]→ R d velocity field or arbitrary control drift depending on context Σ t : [0,T ]→ R d×d diffusion covariance matrix (simplified with scalarσ t throughout the guide) σ t : [0,T ]→ R ≥0 scalar diffusion coefficient c(x,y) :X ×Y → Rused in Section 1 to denote the transport cost c(x,t) : R d × [0,T ]→ Rused in remaining sections to denote state cost or potential V t (x) : R d → R value function defined as the optimal cost-to-go (aim to minimize) ψ t (x) : R d → RLagrange multiplier which defines the value (aim to maximize) Φ(x) : R d → Rterminal cost Auncontrolled generator A u controlled generator ttime coordinate ranging t∈ [0,T ] s another time coordinate, sometimes used to denote the reverse time coordinate s = T − t ranging s∈ [0,T ] when specified τanother time coordinate, typically used for integration ∆tfinite time step θ,φneural network parameters M or M t transport map that maps distributions to new distributions via the push-forward M # p = p ′ 10 1. The Static Schr ̈odinger Bridge Problem We start our deep dive into Schr ̈odinger bridges from its origins in the classical optimal mass transport (OMT) problem in both Monge’s and Kantorovich’s formulations. This perspective naturally frames the static Schr ̈odinger bridge problem as a probabilistic regularization of the OMT problem, linking deterministic transport theory with stochastic processes and entropy minimization 1 . 1.1 The Optimal Mass Transport Problem The origins of the Schr ̈odinger bridge problem can be traced back to the problem of optimal mass transport (OMT) (Villani, 2021), which defines an optimal mapping between points in one distribution to another. In this section, we introduce Monge and Kantorovich’s formulation of the OMT problem, which serves as a starting point for understanding how to optimally transport mass between probability distributions. Consider two probability distributionsπ 0 ∈ P(X) andπ 0 ∈ P(Y) and a transport cost function c(x,y) :X ×Y → Rwhich determines the cost of transporting a unit of mass fromx∈Xtoy ∈Y. We also define the space of transport mapsM:X →Ythat generateπ T as the pushforward ofπ 0 (i.e., M # π 0 = π T ) as: T (π 0 ,π T ) =M :X →Y | M # π 0 = π T (1.1) Among all transport maps, Monge’s OMT problem seeks the optimal mapM ⋆ that minimizes the transport cost. Definition 1.1 (Monge’s Optimal Mass Transport Problem). Given probability distributions π 0 ∈P(X) andπ 0 ∈P(Y) and cost functionc:X ×Y → R, Monge’s OMT problem aims to find the optimal transport map M ⋆ that minimizes: inf M∈T (π 0 ,π T ) Z X c(x,M (x))dπ 0 (x) M :X →Y and M # π 0 = π T (Monge’s OMT Problem) Crucially, (Monge’s OMT Problem) can be an ill-posed problem and yield no solution when the mass of the distributionπ 0 is concentrated at a single point (e.g., a dirac delta) andπ T is concentrated at multiple points (e.g., two dirac deltas with fractional mass), which means mass must be split to reach both targets, and the space of deterministic transport mapsT(π 0 ,π T ) is empty. To avoid the possibility of ill-posedness, Kantorovich defined the OMT problem as an optimization over the space of optimal couplings π 0,T ∈ Π(π 0 ,π T ) defined as: Π(π 0 ,π T ) = π 0,T ∈P(X ×Y) (proj X ) # π 0,T = π 0 , (proj Y ) # π 0,T = π T (1.2) where (proj X ) # π 0,T is theX-marginal ofπ 0,T and vice versa. Π(π 0 ,π T ) is never empty since it always contains the product measure π 0 ⊗ π T . Definition 1.2 (Kantorovich’s Optimal Mass Transport Problem). Given probability distri- butionsπ 0 ∈ P(X) andπ 0 ∈ P(Y) and cost functionc:X ×Y → R, Kantorovich’s OMT problem aims to find the optimal coupling π ⋆ 0,T that minimizes: inf π 0,T Z X×Y c(x,y)dπ 0,T (x,y) π 0,T ∈ Π(π 0 ,π T ) (Kantorovich’s OMT Problem) 1 As the focus of this guide is on Schr ̈odinger bridges for generative modeling, we analyze optimal transport and entropic optimal transport with the intent of connecting it with the dynamic problem in Section 2. For a more comprehensive exposition of entropic OT, we refer the readers to Nutz (2021). 11 Figure 1: Illustration of Relative Entropy or Kullback-Leibler (KL) Divergence. The KL divergence KL(p∥q) between two 1-dimensional probability distributionsp,q ∈P(X) on the state spaceX ⊆ R. The left shows the independent distributions and the right shows the signed contributions ofp log dp dq which yields the KL divergence when integrated, quantifying how well q approximates p. While Kantorovich’s formulation of the OMT problem guarantees existence of an optimal coupling, the resulting problem remains a linear optimization problem inπ 0,T ∈X×Ythat is susceptible to non- unique and deterministic mappings, where for each statex∼ π 0 , the distribution ofπ 0,T (x,·)∈P(Y) is sparse and concentrated at few points. To obtain a smoother and more statistically meaningful coupling, we will consider entropy regularization as a strategy for optimizing a stochastic coupling where each state x∼ π 0 yields a smooth distribution π 0,T (x,·) over possible mappings. 1.2 Entropy on Probability Spaces Before we introduce entropic optimal transport, which leads to the static Schr ̈odinger bridge problem, we need to establish the concept of entropy. Conceptually, entropy measures the uncertainty of one probability measure with respect to another probability measure. We formalize this in the following definition 2 . Definition 1.3 (Entropy Between Probability Measures). Consider two probability measures p,q ∈P(X) on a measureable state spaceX ⊆ R d . The entropy ofprelative to the measure q is defined as: KL(p∥q) := ( E p [log dp dq ] q ≪ p ∞q ̸≪ p (1.3) whereq ≪ pmeans thatqis absolutely continuous with respect topsuch that for all measurable setsA⊆X, we havep(A) = 0 =⇒ q(A) = 0.KL(·∥·) is also known as the Kullback-Leibler (KL) divergence. Intuitively, the Kullback-Leibler (KL) divergenceKL(p∥q) measures the expected log-likelihood ratio underp, which indicates the excess distributional mismatch incurred when data generated under pis modeled underq(Figure 1). In contrast to typical distance measures, the KL divergence is asymmetric, whereKL(p∥q) is a measure of how wellqapproximatesp, which is not equivalent to how wellpapproximatesq. Therefore, we use KL divergence in entropy-regularized OT and Schr ̈odinger bridges to penalize deviations from the reference coupling or path measure. 2 In this section, we usep, ̃p∈P(X) denote arbitrary probability measures onXandq, ̃q ∈P(Y) denote arbitrary probability measures on Y rather than their definitions in the preface. 12 We now establish the chain rule for KL divergence, which decomposes the KL divergence of a joint measure into the divergence between its marginals and the expected conditional divergence. This property allows us to decompose the KL divergence under marginal constraints into a constant and a variable term. Lemma 1.4 (Chain Rule of KL Divergences). Given two joint probability measuresπ X,Y ,π ′ X,Y ∈ P (X ×Y) that are absolutely continuousπ 0,T ≪ π ′ 0,T . Denote theX-marginals asπ X := R Y π 0,T dy andπ ′ X := R Y π ′ 0,T dy and the conditional distribution onYgivenx∈Xasπ Y|X and π ′ Y|X . Then, the KL divergence decomposes into: KL(π X,Y ∥π ′ X,Y ) = KL(π X ∥π Y ) + E x∼π X h KL π Y|X (·|x)∥π ′ Y|X (·|x) i (KL Divergence Chain Rule) Proof. This proof follows simply from the definition of KL divergence: KL(π X,Y ∥π ′ X,Y ) = Z X×Y log dπ X,Y dπ ′ X,Y = Z X×Y log dπ X dπ ′ X dπ Y|X dπ ′ Y|X dπ X,Y = Z X×Y log dπ X dπ ′ X dπ X,Y |z only dependent on x + Z X Z Y log dπ Y|X dπ ′ Y|X dπ Y|X |z =KL(π Y|X ∥π ′ Y|X ) dπ X = Z X log dπ X dπ ′ X dπ X + Z X KL(π Y|X ∥π ′ Y|X )dπ X = KL(π X ∥π ′ X ) + E π X KL(π Y|X ∥π ′ Y|X ) (1.4) which concludes the proof of the KL chain rule.□ The KL chain rule reveals how relative entropy decomposes across joint and conditional distributions. An immediate consequence of this decomposition is the data processing inequality, which formalizes the idea that applying the same stochastic transformation to two probability measures cannot increase their divergence. In other words, when two distributions are passed through the same Markov kernel, the resulting distributions become at least as indistinguishable as the originals. Lemma 1.5 (Data Processing Inequality). Given two probability measuresp,q ∈P(X) and a Markov kernelK:X →P(Y) that maps statesx∈Xto probability measures onY, define ̃q, ̃p∈P(Y) defined as: ̃q(y) := Z X q(x)K(x,y)dx, ̃p(y) := Z X p(x)K(x,y)dx(1.5) Then, they satisfy KL( ̃p∥ ̃q)≤ KL(p∥q). Proof. We prove this by applying the KL chain rule from Lemma 1.4. Let the joint probability measures transformed via the Markov kernel K be denoted as: P (dx,dy) := p(dx)K(x,dy), Q(dx,dy) := p(dx)K(x,dy)(1.6) where ̃p and ̃q are the Y-marginals of P and Q, respectively: ̃p(y) = Z X P (dx,dy)dx, ̃q(y) = Z X Q(dx,dy)dx(1.7) 13 Applying the KL chain rule from Lemma 1.4 to the joint measures, we get: KL(P∥Q) = KL( ̃p∥ ̃q) + E ̃p [KL(P X|Y ∥Q X|Y )] | z ≥0 ≥ KL( ̃p∥ ̃q)(1.8) Since the KL divergence is always non-negative, it follows thatKL( ̃p∥ ̃q)≤ KL(P∥Q). SincePand Qare obtained by applying the same kernelKtopandq, the conditional lawK(x,dy) cancels in the KL divergence as: KL(P∥Q) = Z X×Y log dP dQ (x,y) P (dx,dy) = Z X×Y log p(dx) q(dx) K(x,dy) K(x,dy) | z =1 P (dx,dy) = Z X×Y log dp dq (x) p(dx)K(x,dy) = Z X×Y log dp dq (x) p(dx) Z Y K(x,dy) | z =1 = KL(p∥q) (1.9) Combining (1.8) and (1.9), we have: KL( ̃p∥ ̃q)≤ KL(p∥q)(1.10) which proves the data processing inequality.□ Now that we have developed the notion of entropy on probability spaces as a measure of uncertainty and relative entropy as a measure of distinguishability between distributions, we are ready to introduce the entropic optimal transport (EOT) problem. In this formulation, entropy is used to regularize the classical optimal mass transport problem by penalizing deviation from a reference coupling, yielding a transport objective that admits a stable and unique solution while remaining close, in relative entropy, to a prescribed baseline law. 1.3 Entropic Optimal Transport Problem Having defined the optimal mass transport (OMT) problem as finding a transport plan between marginal distributions that minimizes a cost function, we now extend these ideas to define the entropic optimal transport (EOT) problem (L ́eonard, 2013; Nutz, 2021), which moves closer to the formulation of the Schr ̈odinger bridge problem. Just like the OMT problem, the EOT problem seeks an optimal transport planπ ⋆ 0,T ∈ Π(π 0 ,π T ) between marginalsπ 0 andπ T . However, unlike the OMT problem, optimality is no longer determined solely by the transport costc(x,y) but also an entropy regularization termKL(π 0,T ∥q), where q ∈P(X ×Y) is a reference coupling measure defined on the product space. Definition 1.6 (Entropic Optimal Transport (EOT) Problem). Consider two probability distributionsπ 0 ∈ P(X) andπ T ∈ P(Y) and a cost functionc(x,y) :X ×Y →[0,∞) that defines the cost of transporting units of mass from statesx ∈ Xto statesy ∈ Y. The entropic optimal transport (EOT) problem aims to find the optimal transport planπ ⋆ 0,T that minimizes: inf π 0,T ∈Π(π 0 ,π T ) Z X×Y c(x,y)dπ 0,T (x,y) + αKL(π 0,T ∥q) (Entropic OT Problem) whereKL(·∥·) is the KL divergence (Definition 1.3) between a transport plan and a fixed probability measure q ∈P(X ×Y). From (Entropic OT Problem), we start to observe characteristics of the Schr ̈odinger problem emerge. Specifically, we highlight the KL divergence as a measure of distance from a pre-defined, reference 14 coupling, that prevents the optimal transport coupling from diverging too far from it. This penalty is scaled by a constantα∈ R, where small values result in couplings that allow solutions to diverge more, and large values penalize even small deviations from the reference dynamics. First, we can define the cost functional minimized in (Entropic OT Problem) with the reference measure set to dq := d(π 0 ⊗ π T ) as: F (π 0,T ) := Z X×Y c(x,y)dπ 0,T (x,y) + KL(π 0,T ∥π 0 ⊗ π T )(EOT Functional) Now, we want to define a new reference measure that reduces the (Entropic OT Problem) into a simple KL-minimization problem. To do this, we need a way to absorb the cost integral into the KL minimization objective. Since the reference measure in the KL divergence is defined in the denominator of the logarithm, we scale the product reference measure by the exponential of the negative cost function and some unknown constant α, which transforms into an additive cost when taking the KL. Concretely, we define a tilted reference measure as: d ̃q := e −c α d(π 0 ⊗ π T ), α := Z X×Y d ̃q = Z X×Y e −c d(π 0 ⊗ π T )(1.11) where α is defined such that dq integrates to one. Then, we can expand KL(π 0,T ∥q) as: KL(π 0,T ∥ ̃q) = Z X×Y log dπ 0,T d ̃q dπ 0,T = Z X×Y log dπ 0,T e −c α d(π 0 ⊗ π T ) ! dπ 0,T = Z X×Y log dπ 0,T d(π 0 ⊗ π T ) + log(αe c ) dπ 0,T = Z X×Y log dπ 0,T d(π 0 ⊗ π T ) dπ 0,T | z KL(π 0,T ∥(π 0 ⊗π T )) + Z X×Y c(x,y)dπ 0,T + Z X×Y logαdπ 0,T | z =logα = KL(π 0,T ∥(π 0 ⊗ π T )) + Z X×Y c(x,y)dπ 0,T + logα(1.12) where we separate the integrands and use the fact that integrating a constantlogαover a density returns the constant. Now, we can rearrange to recover an equivalent expression forF(π 0,T ) in (EOT Functional) defined as: KL(π 0,T ∥ ̃q)− logα | z constant = KL(π 0,T ∥(π 0 ⊗ π T )) + Z X×Y c(x,y)dπ 0,T =:F (π 0,T )(1.13) by our definition in (EOT Functional). Sincelogαis a constant independent ofπ 0,T , it does not affect the minimization problem, and the (Entropic OT Problem) reduces to a KL minimization problem: inf π 0,T ∈Π(π 0 ,π T ) Z X×Y c(x,y)dπ 0,T (x,y) + KL(π 0,T ∥q) =inf π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥ ̃q)(1.14) From this result, we can interpret the entropic OT problem as a KL projection of a reference measured ̃qonto the set of couplings with the prescribed marginals Π(π 0 ,π T ). This is exactly the variational structure underlying the static Schr ̈odinger bridge (SB) problem. Before we move on to defining the SB problem, we first consider why we need entropy regularization in the first place over simply minimizing a cost functional, as in the optimal mass transport problem. 15 Motivations for Entropy Regularization Although we have established the definition of entropy and how it can be used to extend the optimal mass transport problem to an entropy-regularized optimal transport problem, a question remains: Why do we need entropy-regularization? The answer can be condensed into the following points: (i)Stochastic Coupling. The entropy term penalizes large probabilities under the couplingπ 0,T which produces a large value forπ 0,T logπ 0,T . While OMT is susceptible to convergence on a singular plan that is concentrated on a lower-dimensional set, such that for each initial pointx 0 ∼ π 0 , all mass is transported to a single or small set of terminal pointsyrather than a smooth density overπ T . Entropy regularization ensures that the solution is a stochastic coupling in which mass is spread smoothly over the joint probability space X ×Y, rather than a deterministic map. (i)Strict Convexity and Uniqueness. The (Monge’s OMT Problem) is linear in π 0,T , such that multiple minimizers can exist with the same total cost. This makes the problem extremely sensitive to initialization, as different initial conditions can yield vastly different solutions. However, since the entropy function defined asπ 0,T 7→ R π 0,T logπ 0,T is convex inπ 0,T , the entropic OT problem has a unique minimizerπ ⋆ 0,T regardless of the initialization. (i)Generalization of Optimal Mass Transport. As the regularization constantε→0, the problem reduces to the optimal mass transport problem. Therefore, the entropic OT problem can be seen as a generalization of the OMT problem with tunable entropic regularization, whereε → ∞makes the cost function negligible, and the solution is equal to the reference coupling q. Now that we have built the foundations of optimal transport of distributions and entropy regularization, we are finally ready to begin our discussion on Schr ̈odinger bridges, starting with the static Schr ̈odinger bridge problem. 1.4 Static Schr ̈odinger Bridge Problem In this section, we will formally define the static Schr ̈odinger bridge (SB) problem, which is closely related to the entropic OT problem discussed in Section 1.3. We introduce the notion of Schr ̈odinger potentials (φ, ˆφ) 3 that uniquely solve a pair of equations, called the Schr ̈odinger system, and simultaneously define the unique static Schr ̈odinger bridge solution. These ideas will form the foundation of our discussion of Sinkhorn’s algorithm in Section 1.5. Definition 1.7 (Static Schr ̈odinger Bridge Problem). Given two marginal distribution con- straintsπ 0 ∈P(X) andπ T ∈P(Y), define the set of all couplings Π(π 0 ,π T )⊂P(X ×Y) with π 0 andπ T as itsX- andY-marginals, respectively. Given a reference measureq ∼ π 0 ⊗ π T a . Then, the static Schr ̈odinger bridge (SB) problem defined as: π ⋆ 0,T =arg min π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥q)(Static SB Problem) 3 Note that in this section and in Section 1.5 we use the notation (φ ⋆ , ˆφ ⋆ ) for the Schr ̈odinger potentials while discussing optimization overφandˆφ. In the remainder of the guide, (φ ⋆ , ˆφ ⋆ ) denotes the unique optimal Schr ̈odinger potentials. 16 where the minimizerπ ⋆ 0,T is unqiue and is called the static Schr ̈odinger bridge betweenπ 0 and π T . Furthermore, when the reference coupling takes the form: q(x,y) := e −c(x,y) α (π 0 ⊗ π T )(x,y)(1.15) The static SB problem coincides with the entropic optimal transport (EOT) problem: π ⋆ 0,T =arg min π 0,T ∈Π(π 0 ,π T ) Z X×Y c(x,y)dπ 0,T (x,y) + αKL(π 0,T ∥q) (1.16) a the symbol∼denotes mutual absolute continuity where forq(S) = 0⇐⇒(π 0 ⊗ π T )(S) = 0 for all measurable sets S ⊆X ×Y The static Schr ̈odinger bridge problem can therefore be viewed as the entropy projection of a reference coupling onto the set of couplings with fixed marginals, and is equivalent to the form of the entropic OT derived in (1.14). Although the static Schr ̈odinger bridge problem is formulated as an optimization over couplings, its solution admits a simple multiplicative structure. In particular, the optimal coupling can be written as a reweighted version of the reference measure using two functions known as the Schr ̈odinger potentials (φ, ˆφ) which together solve the Schr ̈odinger system. Next, we will define the Schr ̈odinger potentials (φ, ˆφ) and show that the pair that solves the Schr ̈odinger system is unique and satisfies the marginal constraints. Proposition 1.8 (Schr ̈odinger Potentials). Consider a reference measureq ≪ π 0 ⊗ π T , which implies that the Radon-Nikodym derivative dq d(π 0 ⊗π T ) is well-defined and given some cost function c :X ×Y → R as: dq d(π 0 ⊗ π T ) = e −c(x,y) (1.17) Then given two functionsφ:X → Randˆφ:Y → R, called the Schr ̈odinger potentials, we define the Schr ̈odinger system as the pair of equations: ( φ(x) =− log R Y e ˆφ(y)−c(x,y) π T (dy) ˆφ(y) =− log R X e φ(x)−c(x,y) π 0 (dx) (Schr ̈odinger System) where the solutionˆπ 0,T solves the static Schr ̈odinger bridge problem (Definition 1.7) and satisfy: dˆπ 0,T (dx,dy) = e φ(x)+ ˆφ(y)−c(x,y) d(π 0 ⊗ π T )(1.18) ˆπ 0 (dx) = π 0 (dx),ˆπ T (dy) = π T (dy)(1.19) whereˆπ 0 (x) := R Y ˆπ 0,T (dx,dy) is the first marginal ofˆπ 0,T andˆπ T (y) := R X ˆπ 0,T (dx,dy) is the second marginal. Furthermore, the potentials (φ, ˆφ) are unique up to an additive constant. Proof. Rather than a direct proof, we present an intuitive derivation of the Schr ̈odinger equations starting from the definition of the static SB problem and its Lagrangian form in Step 1. In Step 2, we prove the uniqueness of (φ, ˆφ). 17 Step 1: Derivation of Schr ̈odinger System from Static SB Problem. First, recall the form static SB problem from Definition 1.7 given by: min ˆπ 0,T ∈Π(π 0 ,π T ) KL(ˆπ 0,T ∥q) =min ˆπ 0,T ∈Π(π 0 ,π T ) Z X×Y log dˆπ 0,T dq dˆπ 0,T (dx,dy), dq := e −c(x,y) d(π 0 ⊗ π T ) (1.20) We can rewrite this objective using Lagrangian multipliersφ(x) andˆφ(y) to enforce the marginal constraints as: L(ˆπ 0,T ) := Z X×Y log dˆπ 0,T dq dˆπ 0,T + Z X φ(x)(π 0 − ˆπ 0 )(dx) + Z Y φ(y)(π T − ˆπ T )(dy)(1.21) where we defineˆπ 0 (x) := R Y ˆπ 0,T (dx,dy) as the first marginal andˆπ T (y) := R X ˆπ 0,T (dx,dy) is the second marginal. To obtain the optimality conditions, we take the functional derivative ofL(ˆπ 0,T ) with respect to ˆπ 0,T and set the first variation to zero to get: δ δ ˆπ 0,T L(ˆπ 0,T ) = log dˆπ 0,T dq + 1− φ(x)− ˆφ(y) = 0 =⇒ log dˆπ 0,T dq =−1 + φ(x) + ˆφ(y) =⇒ dˆπ 0,T dq = e −1 e φ(x)+ ˆφ(y) =⇒ dˆπ 0,T (dx,dy) = e φ(x)+ ˆφ(y)−c(x,y) d(π 0 ⊗ π T ) = e φ(x)+ ˆφ(y)−c(x,y) π 0 (dx)π T (dy)(1.22) where the constante −1 can be absorbed into the Lagrange multipliers. Now, we can compute the expression for each marginal of ˆπ 0,T by integration: ˆπ 0 (dx) = Z Y dˆπ 0,T (dx,dy) = Z Y e φ(x)+ ˆφ(y)−c(x,y) π 0 (dx)π T (dy) = e φ(x) π 0 (dx) Z Y e ˆφ(y)−c(x,y) π T (dy) =⇒ e φ(x) π 0 (dx) Z Y e ˆφ(y)−c(x,y) π T (dy) = π 0 (dx) =⇒ log e φ(x) Z Y e ˆφ(y)−c(x,y) π T (dy) = log(1) =⇒ φ(x) =− log Z Y e ˆφ(y)−c(x,y) π T (dy)(1.23) Similarly, for the second marginal, we have: ˆπ T (dy) = Z X dˆπ 0,T (x,y) = Z X e φ(x)+ ˆφ(y)−c(x,y) π 0 (dx)π T (dy) = e ˆφ(y) π T (dy) Z X e φ(x)−c(x,y) π 0 (dx) =⇒ e ˆφ(y) π T (dy) Z X e φ(y)−c(x,y) π 0 (dx) = π T (dy) =⇒ log e ˆφ(x) Z X e φ(x)−c(x,y) π 0 (dx) = log(1) =⇒ ˆφ(y) =− log Z Y e φ(x)−c(x,y) π 0 (dx)(1.24) Together, (1.23) and (1.24) recover the (Schr ̈odinger System) and establishes the proof that the Schr ̈odinger equation forφ(x) enforces the first marginal constraintˆπ 0 =π 0 and the Schr ̈odinger equation for ˆφ(x) enforces the second marginal constraint ˆπ T = π T . Step 2: Proving Uniqueness of Schr ̈odinger Potentials. From our derivation in Step 1, we show that the solution to the Lagrangian form of the static SB problem can be written as: ∀(x,y)∈X ×Y, dˆπ 0,T dq = e φ(x)+ ˆφ(y) =⇒ log dˆπ 0,T dq = φ(x) + ˆφ(y)(1.25) To prove that (φ, ˆφ) are unique for the solutionˆπ 0,T , consider two new functionsφ ′ :X → Rand ˆφ ′ :Y → R which also satisfy: log dˆπ 0,T dq = φ(x) + ˆφ(y) = φ ′ (x) + ˆφ ′ (y) =⇒ φ(x)− φ ′ (x) = ˆφ ′ (y)− ˆφ(y), q-a.s.(1.26) 18 Since the identity holds forq-a.s. 4 onX ×Y, by Fubini’s theorem 5 , there existx ⋆ ∈Xandy ⋆ ∈Y, such that the identity (1.26) holds forπ 0 -a.e.(almost everywhere) inxfor fixedy=y ⋆ andπ T -a.e. inyfor fixedx=x ⋆ . Since the left-hand side is dependent only onxand the right-hand side is dependent only on y, fixing y = y ⋆ and defining a := ˆφ ′ (y ⋆ )− ˆφ(y ⋆ ) yields: φ(x)− φ ′ (x) = ˆφ ′ (y ⋆ )− ˆφ(y ⋆ ) =: a, π 0 -a.s.(1.27) =⇒ ( φ(x) = φ ′ (x) + a π 0 -a.s. ˆφ(y) = ˆφ ′ (y)− a π T -a.s. (1.28) which proves that (φ, ˆφ) are unique up to a constant a∈ R.□ Using these results, we can show that the Schr ̈odinger potentials (φ, ˆφ) also define the unique optimal coupling π ⋆ 0,T that solves (Static SB Problem). Proposition 1.9 (Solution to Static SB Problem). Given two marginalsπ 0 ∈ P(X) and π T ∈P(Y) and a reference couplingq ∼ π 0 ⊗ π T , assume the set of finite entropy couplings Π(π 0 ,π T )̸=∅. Let ˆπ 0,T be a coupling that satisfies: log dˆπ 0,T dq = φ⊕ ˆφ, q-a.s.(1.29) for measurable functionsφ:X → Randˆφ:Y → R. Thenˆπ 0,T =π ⋆ 0,T that solves (Static SB Problem) and yields a constant map: π 0,T 7→ E π 0,T log dπ ⋆ 0,T dq , ∀π 0,T ∈ Π(π 0 ,π T )∪π ⋆ 0,T (1.30) Proof. Step 1: Prove the Constant Map. Assumeφ∈ L 1 (π 0 ) andˆφ∈ L 1 (π T ) are measurable and integrable. Consider the optimal coupling π ⋆ 0,T which satisfies: log dπ ⋆ 0,T dq = φ⊕ ˆφ := φ(x) + ˆφ(y), q-a.s.(1.31) Given (1.31) and absolute continuity π 0,T ≪ q, we can write: E π 0,T log dπ ⋆ 0,T dq = E π 0,T [φ(x) + ˆφ(y)] = Z X×Y φ(x)π 0,T (dx,dy) + Z X×Y ˆφ(y)π 0,T (dx,dy) = Z X φ(x)π 0 (dx) + Z Y ˆφ(y)π T (dy) | z independent on coupling π 0,T (1.32) where the last equality follows from the fact thatπ 0,T has marginalsπ 0 andπ T . Since the final equality is only dependent on π 0 and π T and not the coupling π 0,T , we have that the map: π 0,T 7→ E π 0,T log dπ ⋆ 0,T dq (1.33) is constant over Π(π 0 ,π T ). Step 2: Prove Optimality. Now, we aim to show thatπ ⋆ 0,T uniquely minimizesKL(π 0,T ∥q) over Π. For any couplingπ 0,T ∈Π(π 0 ,π T ), we can decompose the KL divergenceKL(π 0,T ∥q) with respect to 4 holds for all (x,y)∈X ×Y outside the null set where q(N ) = 0 5 which states that integrating over the product space is equal to integrating over each space sequentially 19 π ⋆ 0,T as: KL(π 0,T ∥q) = E π 0,T log dπ 0,T dq = E π 0,T " log dπ 0,T dπ ⋆ 0,T # |z =KL(π 0,T ∥π ⋆ 0,T ) +E π 0,T log dπ ⋆ 0,T dq |z constant in π 0,T given (1.33) = KL(π 0,T ∥π ⋆ 0,T ) + C(1.34) where we use the constant map property from Step 1 (1.33). Since the KL divergence is non-negative with equalityKL(π 0,T ∥π ⋆ 0,T ) = 0 if and only ifπ 0,T =π ⋆ 0,T , we have thatπ ⋆ 0,T is the unqiue solution to (Static SB Problem).□ From this proof, we observe that (φ, ˆφ) depend only on the marginal constraints and thus act as Lagrange multipliers on the marginal constraints. This naturally leads to an alternative dual formulation of the static SB problem, which aims to maximize the potentials to enforce the marginal constraints. Theorem 1.10 (Dual Formulation of Static SB Problem). The equivalent dual formulation of (Static SB Problem) is defined as: inf π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥q) |z primal problem = sup φ, ˆφ Z X φdπ 0 + Z Y ˆφdπ T − Z X×Y e φ⊕ ˆφ dq + 1 |z dual problem (Strong Duality) where the supremum is achieved by the Schr ̈odinger potentials (φ ⋆ , ˆφ ⋆ ) which define associated Schr ̈odinger bridge coupling π ⋆ 0,T and satisfy: KL(π ⋆ 0,T ∥q) =inf π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥q) = Z X φ ⋆ dπ 0 + Z Y ˆφ ⋆ dπ T , dπ ⋆ 0,T dq = e φ ⋆ ⊕ ˆφ ⋆ q-a.s. (1.35) where φ ⋆ ∈ L 1 (π 0 ), ˆφ ⋆ ∈ L 1 (π T ) and φ ⋆ ⊕ ˆφ ⋆ is unique. Proof. First, we will prove weak duality using Fenchel’s inequality before we prove strong duality, which establishes the relationship between the optimal Schr ̈odinger potentials (φ ⋆ , ˆφ ⋆ ) and the Schr ̈odinger bridge coupling π ⋆ 0,T . Step 1: Prove Weak Duality. Weak duality states that for any measurable functionsφ∈ L 1 (π 0 ), ˆφ∈ L 1 (π T ) and coupling π 0,T ∈ Π(π 0 ,π T ), the following inequality holds: KL(π 0,T ∥q)≥ Z X φdπ 0 + Z Y ˆφdπ T − Z X×Y e φ⊕ ˆφ dq + 1(Weak Duality) First, we define the Fenchel’s inequality which states that for any convex functionfand anyα,β, the inequality is satisfied: f (α)≥ αβ− f ⋆ (β),where f ⋆ (β) := sup α≥0 αβ− f (α)(1.36) wheref ⋆ (β) is the Fenchel conjugate off(α) which defines all supporting affine functions forf(α) 6 . Since the KL divergence can be written as an integral overα logα, we define the convex function f(α) =α logα− αwhich yields the clean convex conjugatef ⋆ (α) =sup α≥0 αβ− α logα+α=e β , 6 the expressionαβ− f(α) arises from defining any affine functionαβ− γ ≤ f(α),∀αthat lies below the convex function at all pointsα, rearranging to getγ ≥ αβ− h(α),∀α, and defining the Fenchel conjugate as the smallest possible c to get f ⋆ (α) := sup α≥0 αβ− f (α) 20 which can be derived by computing the derivative ofαβ− α logα+αand setting it to zero 7 . Then, by Fenchel’s inequality, we have: α logα− α≥ βα− e β =⇒ α logα− βα≥ α− e β , ∀α≥ 0,β ∈ R(1.37) By settingα:= dπ 0,T dq andβ:=φ⊕ ˆφ, we see thatα logαrecovers the KL divergence ande β becomes e φ⊕ ˆφ . Then, we can expand the KL divergence as: KL(π 0,T ∥q) = Z X×Y dπ 0,T dq log dπ 0,T dq dq = Z X×Y dπ 0,T dq log dπ 0,T dq dq+ Z X×Y (φ⊕ ˆφ) dπ 0,T dq dq− Z X×Y (φ⊕ ˆφ) dπ 0,T dq dq = Z X×Y (φ⊕ ˆφ) dπ 0,T dq dq + Z X×Y dπ 0,T dq log dπ 0,T dq − (φ⊕ ˆφ) dπ 0,T dq | z =:α logα−βα dq ≥ Z X×Y (φ⊕ ˆφ)dπ 0,T + Z X×Y dπ 0,T dq − e φ⊕ ˆφ dq ≥ Z X φdπ 0 + Z Y ˆφdπ T + Z X×Y dπ 0,T | z =1 − Z X×Y e φ⊕ ˆφ dq ≥ Z X φdπ 0 + Z Y ˆφdπ T − Z X×Y e φ⊕ ˆφ dq + 1(1.38) which concludes our proof of weak duality. Step 2: Prove Strong Duality. To prove strong duality, we recall the definition of the optimal Schr ̈odinger potentials (φ ⋆ , ˆφ ⋆ ) and the optimal Schr ̈odinger bridge couplingπ ⋆ 0,T ∈ Π(π 0 ,π T ) which satisfy: dπ ⋆ 0,T dq = e φ ⋆ ⊕ ˆφ ⋆ q-a.s. where φ ⋆ ∈ L 1 (π 0 ), ˆφ∈ L 1 (π T )(1.39) Given the inequality in (Weak Duality), we can take the infimum on the left-hand side and the supremum on the right-hand side, which preserves the inequality to get: inf π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥q)≥sup φ∈L 1 (π 0 ), ˆφ∈L 1 (π T ) Z X φdπ 0 + Z Y ˆφdπ T − Z X×Y e φ⊕ ˆφ dq + 1 |z =:G(φ, ˆφ) (1.40) where we letG(φ, ˆφ) define the dual objective. Now, we will expand both sides of the inequality for the optimalφ ⋆ , ˆφ ⋆ ,π ⋆ 0,T . For the left-hand side, we apply (1.39) to derive the KL divergence for the optimal coupling π ⋆ 0,T as: KL(π ⋆ 0,T ∥q) = Z X×Y log dπ ⋆ 0,T dq dπ ⋆ 0,T = Z X×Y (φ ⋆ ⊕ ˆφ)dπ ⋆ 0,T = Z X φ ⋆ dπ ⋆ 0,T + Z Y ˆφ ⋆ dπ ⋆ 0,T = Z X φ ⋆ dπ 0 + Z Y ˆφ ⋆ dπ T (1.41) 7 explicitly, we letg(α) :=αβ− α logα+α, take the derivativeg ′ (α) =β− logα−1 + 1, and set to zero to get β = logα =⇒ α = e β , which is the supremum since f is convex. 21 Figure 2: Illustration of Sinkhorn’s Algorithm. Starting from an initial potential (e.g.φ 0 := 0), Sinkhorn’s algorithm alternates updates of the dual potentialsφ n andˆφ n via log-integral transforms involving the transport costc(x,y) and the marginal constraintsπ 0 andπ T . Each alternating step enforces one marginal constraint while preserving the entropic structure, and the sequence (φ n , ˆφ n ) converges to the optimal dual pair (φ ⋆ , ˆφ ⋆ ) that uniquely defines the static Schr ̈odinger bridge coupling π ⋆ 0,T . For the right-hand side, we use the definition dπ ⋆ 0,T = e φ ⋆ ⊕ ˆφ dq to get: G(φ ⋆ , ˆφ ⋆ ) = Z X φdπ 0 + Z Y ˆφdπ T − Z X×Y e φ⊕ ˆφ dq + 1 = Z X φdπ 0 + Z Y ˆφdπ T − Z X×Y dπ ⋆ 0,T | z =1 + 1 = Z X φdπ 0 + Z Y ˆφdπ T (1.42) Since (1.41) and (1.42) are equal for the optimalφ ⋆ , ˆφ ⋆ ,π ⋆ 0,T , it follows that the infimum isKL(π ⋆ 0,T ∥q) and equal to the supremum at optimality: inf π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥q) = KL(π ⋆ 0,T ∥q) = G(φ ⋆ , ˆφ ⋆ ) =sup φ∈L 1 (π 0 ), ˆφ∈L 1 (π T ) G(φ, ˆφ)(1.43) which proves (Strong Duality) and establishes that equality between objectives occurs exactly at the Schr ̈odinger bridge solutionφ ⋆ , ˆφ ⋆ ,π ⋆ 0,T . Since−e φ⊕ ˆφ is strictly concave in (φ⊕ ˆφ), the sum (φ ⋆ ⊕ ˆφ ⋆ ) is the unique maximizer, which also implies thatφandˆφare unique up to a constant.□ In this section, we reformulated the static SB problem from the problem of optimizing an optimal transport couplingπ ⋆ 0,T on the high-dimensional product spaceX×Yto a problem of determining two scalar SB potential functions (φ, ˆφ) defined on the marginalsπ 0 andπ T which uniquely characterize the optimal bridge. Leveraging this simplified structure, we can now turn to establishing a tractable algorithm for solving the static SB problem. 1.5 Sinkhorn’s Algorithm We now introduce the classical algorithm used to solve the static Schr ̈odinger system defined in (Schr ̈odinger System), known as Sinkhorn’s algorithm in optimal transport (Cuturi, 2013; Knight, 2008; Nutz, 2021; Sinkhorn, 1967) or the Iterative Proportional Fitting (IPF) procedure in statistics (Fortet, 1940; Kullback, 1968; R ̈uschendorf, 1995). 22 Recall that (Schr ̈odinger System) consists of two equations for the pair of Schr ̈odinger potentials (φ, ˆφ) defined as: φ(x) =− log Z Y e ˆφ(y)−c(x,y) π T (dy)(First Potential) ˆφ(y) =− log Z X e φ(x)−c(x,y) π 0 (dx)(Second Potential) Optimizing bothφandˆφsimultaneously would lead to a mismatch, as they depend on each other. Therefore, it is natural to consider an alternating optimization scheme that optimizes one of the potentials with the other potential fixed. This is exactly the intuition behind Sinkhorn’s algorithm (Figure 2). The algorithm starts by initializing the first potential at someφ=φ 0 and defines an alternating optimization sequenceφ n , ˆφ n n≥0 . We can also consider this alternating sequence as maximizing the dual problem defined in (Strong Duality) with the objective: G(φ, ˆφ) := Z X φdπ 0 + Z Y ˆφdπ T − Z X×Y e φ⊕ ˆφ dq + 1(Dual Objective) Combining both the Schr ̈odinger system equations and the dual problem, we define the alternating optimization sequence φ n , ˆφ n n≥0 with φ 0 := 0 as: (i) Solveˆφ n using (Second Potential) withφ:=φ n . Equivalently, solveˆφ n :=arg maxG(φ n ,·) using (Dual Objective). (i) Solveφ n+1 using (First Potential) withˆφ:=ˆφ n . Equivalently, solveφ n+1 :=arg maxG(·, ˆφ n ) using (Dual Objective). Since (Dual Objective) is strictly concave with respect to bothφandˆφ, each iteration strictly increases the objectiveG(φ n , ˆφ n )< G(φ n+1 , ˆφ n )< G(φ n+1 , ˆφ n+1 ), unless the optimal pair (φ ⋆ , ˆφ ⋆ ) is reached, whereG(φ n , ˆφ n ) =G(φ n+1 , ˆφ n ) =G(φ n+1 , ˆφ n+1 ). Furthermore, the coupling at each iteration is defined as: dπ 0,T (φ, ˆφ) := e φ⊕ ˆφ dq = e φ⊕ ˆφ−c d(π 0 ⊗ π T ) define: π 2n 0,T := π 0,T (φ n , ˆφ n ) π 2n−1 0,T := π 0,T (φ n , ˆφ n−1 ) π −1 0,T = π 0,T (0, 0) = e −c d(π 0 ⊗ π T ) =: dq (1.44) Given our definition above, we can now define a few properties of Sinkhorn’s algorithm which will aid us in proving its convergence. Lemma 1.11 (Properties of Sinkhorn’s Algorithm). The sequence of potentialsφ n , ˆφ n n≥0 , whereφ n ∈ L 1 (π 0 ) andˆφ n ∈ L 1 (π T ) are integrable, and coupled densitiesπ 2n 0,T ,π 2n−1 0,T n≥0 defined by the Sinkhorn iterations satisfy the following properties: (i) Each KL step equals a difference of potentials: KL π 2n 0,T ∥π 2n−1 0,T = Z Y ( ˆφ n − ˆφ n−1 )π T (dy),KL π 2n+1 0,T ∥π 2n 0,T = Z X (φ n+1 − φ n )π 0 (dx) (i) The total dual potential equals the total accumulated KL cost: π T ( ˆφ n ) = n X k=0 KL π (2k) 0,T ∥π (2k−1) 0,T , π 0 (φ n ) = n−1 X k=0 KL π (2k+1) 0,T ∥π (2k) 0,T 23 where π 0 (φ n ) and π T ( ˆφ n ) are non-negative and increasing. Proof. We will prove each part of the Lemma in steps. Step 1: Proof of Property (i). To prove this, we start with the definition of the KL divergence: KL π 2n 0,T ∥π 2n−1 0,T = Z X×Y log dπ 2n 0,T dπ 2n−1 0,T ! dπ 2n 0,T ,KL π 2n+1 0,T ∥π 2n 0,T = Z X×Y log dπ 2n+1 0,T dπ 2n 0,T ! dπ 2n+1 0,T (1.45) From (1.44), we can decompose the log ratio of the density updates as: dπ 2n 0,T dπ 2n−1 0,T = e φ n (x)+ ˆφ n (y)−c(x,y) d(π 0 ⊗ π T ) e φ n (x)+ ˆφ n−1 (y)−c(x,y) d(π 0 ⊗ π T ) = e ˆφ n (y) e ˆφ n−1 (y) = e ˆφ n (y)− ˆφ n−1 (y) (1.46) dπ 2n+1 0,T dπ 2n 0,T = e φ n+1 (x)+ ˆφ n (y)−c(x,y) d(π 0 ⊗ π T ) e φ n (x)+ ˆφ n (y)−c(x,y) d(π 0 ⊗ π T ) = e φ n+1 (x) e φ n (x) = e φ n+1 (x)−φ n (x) (1.47) Plugging the result (1.46) into (1.45), we get: KL π 2n 0,T ∥π 2n−1 0,T = Z X×Y log e ˆφ n (y)− ˆφ n−1 (y) dπ 2n 0,T = Z X×Y ( ˆφ n (y)− ˆφ n−1 (y))dπ 2n 0,T = Z Y ( ˆφ n (y)− ˆφ n−1 (y))π T (dy)(1.48) Similarly, plugging in (1.47) into (1.45), we have: KL π 2n+1 0,T ∥π 2n 0,T = Z X×Y log e φ n+1 (x)−φ n (x) dπ 2n+1 0,T = Z X×Y (φ n+1 (x)− φ n (x))dπ 2n+1 0,T = Z X (φ n+1 (x)− φ n (x))π 0 (dx)(1.49) which recovers the KL steps defined in (i). Since the KL iterates are finite, we can conclude that the potentials are integrable over the marginal densities such thatφ n ∈ L 1 (π 0 ) andˆφ n ∈ L 1 (π T ). In the next step, we show that the Sinkhorn iterates admit an additive structure, where the total dual potentials are equivalent to the sum of KL divergences between iterations. Step 2: Proof of Property (i). We can sum up the KL divergence expressions derived in (1.48) and (1.49) to recover the total accumulated KL at each iteration: n X k=0 KL π (2k) 0,T ∥π (2k−1) 0,T = n X k=0 Z Y ( ˆφ k − ˆφ k−1 )π T = Z Y n X k=0 ( ˆφ k − ˆφ k−1 ) | z =( ˆφ n − ˆφ −1 ) π T (⋆) = Z Y ˆφ n π T =: π T ( ˆφ n ) (1.50) n−1 X k=0 KL π (2k+1) 0,T ∥π (2k) 0,T = n−1 X k=0 Z X (φ k+1 − φ k )π 0 = Z X n−1 X k=0 (φ k+1 − φ k ) |z =(φ n −φ 0 ) π 0 (⋆) = Z X φ n π 0 =: π 0 (φ n )(1.51) where the equalities (⋆) follow from applying the telescoping trick to cancel intermediate terms and substituting the initialization ofˆφ −1 = 0 andφ 0 = 0. This recovers a clean expression for the total accumulated KL after n iterations of Sinkhorn’s algorithm.□ To show convergence of Sinkhorn iterations, we aim to show that the KL divergence with the optimal Schr ̈odinger bridge coupling converges to zero, i.e.,KL(π ⋆ 0,T ∥π (n) 0,T )→0, and achieves the correct marginalsπ 0 andπ T asn→∞iterations, which implies that the potentials converge to the Schr ̈odinger potentials (φ, ˆφ)→ (φ ⋆ , ˆφ ⋆ ). 24 Proposition 1.12 (Marginal Convergence of Sinkhorn Iterations). Each iteration of Sinkhorn’s algorithm results in a decrease in KL divergence, such that for alln≥−1, the KL with the optimal coupling π ⋆ 0,T is decreasing in n: KL π ⋆ 0,T ∥π (n) 0,T = KL π ⋆ 0,T ∥q − n X k=0 KL π (k) 0,T ∥π (k−1) 0,T (1.52) Proof. Using the properties of Sinkhorn iterations defined in Lemma 1.11, we can expand the expression for the KL divergence with the optimal coupling π ⋆ 0,T as: KL(π ⋆ 0,T ∥π 2n 0,T ) = E π ⋆ 0,T " log dπ ⋆ 0,T dπ 2n 0,T # = E π ⋆ 0,T " log dπ ⋆ 0,T dq dq dπ 2n 0,T # = E π ⋆ 0,T log dπ ⋆ 0,T dq | z KL(π ⋆ 0,T ∥q) −E π ⋆ 0,T log dπ 2n 0,T dq = KL(π ⋆ 0,T ∥q)− E π ⋆ 0,T " n X k=0 log dπ (2k) 0,T dπ (2k−1) 0,T + n−1 X k=0 log dπ (2k+1) 0,T dπ (2k) 0,T # = KL(π ⋆ 0,T ∥q)− E π ⋆ 0,T " n X k=0 ( ˆφ k − ˆφ k−1 ) + n−1 X k=0 (φ k+1 − φ k ) # = KL(π ⋆ 0,T ∥q)− E π ⋆ 0,T [( ˆφ n − ˆφ −1 ) + (φ n − φ 0 )] = KL(π ⋆ 0,T ∥q)− E π ⋆ 0,T [ ˆφ n + φ n ] = KL(π ⋆ 0,T ∥q)− (π T ( ˆφ n ) + π 0 (φ n ))(1.53) Substituting the result from Lemma 1.11 (i), we have: KL(π ⋆ 0,T ∥π 2n 0,T ) = KL(π ⋆ 0,T ∥q)− (π T ( ˆφ n ) + π 0 (φ n )) = KL(π ⋆ 0,T ∥q)− n X k=0 KL π (2k) 0,T ∥π (2k−1) 0,T + n−1 X k=0 KL π (2k+1) 0,T ∥π (2k) 0,T ! = KL(π ⋆ 0,T ∥q)− 2n X k=0 KL π (k) 0,T ∥π (k−1) 0,T =⇒ KL(π ⋆ 0,T ∥π (n) 0,T ) = KL(π ⋆ 0,T ∥q)− n X k=0 KL π (k) 0,T ∥π (k−1) 0,T (1.54) which proves that the KL divergence with the optimal coupling is decreasing as the number of iterations n increases.□ We can now establish the convergence of Sinkhorn’s algorithm, which states that as the number of iterations increases to infinityn→∞, the KL divergence with the optimal coupling converges to zero. Corollary 1.13 (Convergence of Sinkhorn’s Algorithm). The KL divergence between the marginals satisfies the inequality: KL(π (k) 0 ∥π 0 ) + KL(π (k) T ∥π T )≤ KL(π (k) 0,T ∥π (k−1) 0,T )(1.55) 25 and the sum of the right-hand side for n iterations is bounded by: n X k=1 KL(π (k) 0,T ∥π (k−1) 0,T )≤ KL(π ⋆ 0,T ∥q)(1.56) =⇒ KL(π (k) 0,T ∥π (k−1) 0,T )→ 0, KL(π (k) 0 ∥π 0 )→ 0,KL(π (k) T ∥π T )→ 0,for n→∞(1.57) which also implies that π (k) 0 → π 0 and π (k) T → π T as n→∞. Proof. First, we recall that the marginals are correct forπ 0 on the odd iterationsk ≥1, such that π (k) 0 =π 0 , and correct forπ T for the even iterationsk ≥2, such thatπ (k) T =π T . Therefore, we can rewrite the sum of the marginal KL divergences as: KL(π (2k) 0 ∥π 0 ) + KL(π (2k) T ∥π T ) |z =0 = KL(π (2k) 0 ∥π 0 )≤ KL(π (2k) 0,T ∥π (2k−1) 0,T )(1.58) KL(π (2k−1) 0 ∥π 0 ) | z =0 +KL(π (2k−1) T ∥π T ) = KL(π (2k−1) T ∥π T )≤ KL(π (2k−1) 0,T ∥π (2k−2) 0,T )(1.59) where the inequalities can be obtained by applying the data processing inequality (Lemma 1.5). Forniterations, we can rearrange (1.52) to get an upper bound for the sum of KL divergences for k ≥ 1 as: n X k=0 KL π (k) 0,T ∥π (k−1) 0,T = KL π ⋆ 0,T ∥q − KL π ⋆ 0,T ∥π (n) 0,T |z ≥0 ≤ KL π ⋆ 0,T ∥q (1.60) =⇒ n X k=1 KL π (k) 0,T ∥π (k−1) 0,T = n X k=0 KL π (k) 0,T ∥π (k−1) 0,T − KL π (0) 0,T ∥π (−1) 0,T ≤ KL π ⋆ 0,T ∥q − KL π (0) 0,T ∥q (1.61) where π (−1) 0,T = dq as in (1.44). Therefore, for n→∞, we have: ∞ X k=1 KL(π (2k) 0 ∥π 0 )≤ ∞ X k=1 KL π (k) 0,T ∥π (k−1) 0,T ≤ KL π ⋆ 0,T ∥q − KL π (0) 0,T ∥q =⇒ KL(π (n) 0 ∥π 0 )→ 0 (1.62) ∞ X k=1 KL(π (2k−1) T ∥π T )≤ ∞ X k=1 KL π (k) 0,T ∥π (k−1) 0,T ≤ KL π ⋆ 0,T ∥q − KL π (0) 0,T ∥q =⇒ KL(π (n) T ∥π T )→ 0 (1.63) Applying Pinkser’s inequality, which states that the total variation between distributionsp,qsatisfy ∥p− q∥ TV ≤ p 2KL(p∥q), we conclude that π (n) 0 → π 0 and π (n) T → π T in variation.□ While Proposition 1.12 proves that the marginals generated by the Sinkhorn iterations converge to the marginal constraints, it does not directly imply strong convergence to the optimal couplingπ ⋆ 0,T or the Schr ̈odinger potentials (φ, ˆφ)→(φ ⋆ , ˆφ ⋆ ). However, strong convergence is only guaranteed for certain conditions on c, which we establish with the following Theorem. Theorem 1.14 (Strong Convergence of Sinkhorn’s Algorithm). Given a cost functionc: X ×Y → R that is bounded from below and satisfies exponential integrability, such that: ∃r > 1, Z e rc(x,y) d(π 0 ⊗ π T ) <∞(Integrability Condition) 26 Then, the Sinkhorn iterates converge to the true Schr ̈odinger potentialsφ n → φ ⋆ andˆφ n → ˆφ ⋆ , and the induced couplings converge to the optimal Schr ̈odinger bridge,KL(π ⋆ 0,T ∥π (n) 0,T )→0 and π (n) 0,T → π ⋆ 0,T in variation. Proof Sketch. Under the (Integrability Condition) on the cost, the Sinkhorn iterates remain uniformly integrable and cannot develop singular behavior. Since the iterates are uniformly integrable, functional analysis establishes that there exists a subsequence wheree φ n converges weakly inL 1 , which passes the limit and forces convergence of ˆφ n → ˆφ ⋆ under the coupled update condition: Z X e φ n (x) e −c(x,y) dπ 0 (x)→ Z X e φ ⋆ (x) e −c(x,y) dπ 0 (x) |z converges weakly in L 1 (π 0 ) =⇒ − log lim n→∞ Z X e φ n (x) e −c(x,y) dπ 0 (x) = − log Z X e φ ⋆ (x) e −c(x,y) dπ 0 (x) =: ˆφ ⋆ (y) (1.64) We have shown in Theorem 1.10 that (φ⊕ ˆφ) is unique, which implies that all subsequences of iterations converge to the same limit. Therefore, the full sequence of Sinkhorn iterates converges, and the induced couplings converge in KL and total variation to the Schr ̈odinger bridge 8 . □ We have now established the first method to tractably solve the Schr ̈odinger bridge problem under the static formulation, which is equivalent to solving the entropic optimal transport problem. From our analysis, we have shown that the intuitive idea of alternating between solving (First Potential) and (Second Potential) leads to a theoretically grounded algorithm which establishes Sinkhorn iterations as alternating KL projections onto the marginal constraint sets, or equivalently as coordinate ascent on the dual Schr ̈odinger potentials. These ideas lay the foundation of computational optimal transport and modern generative modeling techniques. 1.6 Closing Remarks for Section 1 In this section, we traced the origins of the Schr ̈odinger bridge problem to the classical optimal mass transport (OMT) problem introduced by Monge and Kantorovich, which seeks an optimal coupling between probability distributions that minimizes a prescribed transport cost. Motivated by the non-uniqueness and instability that can arise in the solutions of the OMT problem, we introduced the formulation of entropic optimal transport (EOT), in which an entropy-regularization term that measures deviation from a reference coupling is added to the transport objective. The resulting entropy- regularized problem admits a unique and computationally stable solution. By reparameterizing the reference coupling, we showed that the EOT problem is equivalent to the static Schr ̈odinger bridge (SB) problem, where the objective is to find a coupling that remains closest in relative entropy to a reference endpoint law while satisfying the prescribed marginal distributions. Finally, we introduced the classical Sinkhorn algorithm, an efficient iterative scheme for solving the static SB problem in practice. While the static formulation provides the foundation for understanding how probability mass can be optimally transported between distributions, its connection to modern generative modeling emerges through the dynamic formulation of the Schr ̈odinger bridge problem. The dynamic viewpoint lifts the static coupling problem to the space of stochastic path measures, describing how probability flows through time under controlled stochastic dynamics. In the next section, we develop this perspective by introducing the theory of stochastic processes and path measures, defining KL divergences over path space, and deriving the optimality conditions of the dynamic Schr ̈odinger bridge in the form of both nonlinear and linear partial differential equations (PDEs). 8 For rigorous proof, see Theorem 6.15 in Nutz (2021). 27 2. The Dynamic Schr ̈odinger Bridge Problem Having established the connection between optimal transport and entropy-regularized couplings, we now turn to the dynamic formulation of the Schr ̈odinger bridge (SB) problem. While optimal transport focuses on static couplings between distributions, the dynamic SB lifts the problem to the level of stochastic processes. Instead of directly transporting mass, we seek the most likely evolution of a stochastic system that transforms an initial distribution into a target distribution over time. In this section, we formalize this dynamic viewpoint and develop the stochastic calculus tools, including path measures, Itˆo processes, and change-of-measure techniques, necessary to analyze and solve the dynamic SB problem. 2.1 Dynamic Optimal Transport Problem Just like how the static Schr ̈odinger bridge problem could be traced back to the Monge-Kantorovich optimal mass transport problems, we begin our discussion of the dynamic Schr ̈odinger bridge problem with Bernamou-Brenier (dynamic) optimal transport problem (Benamou et al., 2000). The key idea that differentiates the dynamic from the static problems is the intuition behind transporting mass with a continuous flow over a time interval rather than static couplings between marginals. Concretely, given an initial distributionπ 0 ∈P(R d ) at timet= 0 and a target distributionπ T ∈P(R d ) at timet=T, the dynamic formulation aims to determine the continuous-time evolution of a marginal probability density p t ∈P(R d ) that evolves mass from π 0 → π T over t∈ [0,T ]. Recall that (Kantorovich’s OMT Problem) aims to minimize the cost of transporting statesx 0 ∼ π 0 tox T ∼ π T defined asc(x 0 ,x T ). A natural cost function is the Euclidean distancec(x 0 ,x T ) := ∥x T − x 0 ∥ 2 , which is the straight-line distance betweenx 0 andx T . To reformulate this into a dynamic transport problem, we can consider transporting the state continuous statexover the straight line which travels a total distance of∥x T −x 0 ∥ 2 . Concretely, the quadratic transport cost fromx 0 → x T is equivalent to the integrating energy-minimizing velocity fieldv t (x) := d dt x t over t∈ [0,T ] defined as: ∥x T −x 0 ∥ 2 =inf x t :x 0 →x T Z T 0 ∥v t (x)∥ 2 dt(2.1) Leveraging this identity, we can reframe the static OT problem as minimizing the total kinetic energy of transporting particles from the distributionπ 0 , where particles at locationxmove with velocityv t (x), weighted by the probability mass of the particlep t (x) (Figure 3). Jointly minimizing the velocity and marginal probability path (v t ,p t ) yields the dynamic OT problem. Definition 2.1 (Dynamic Optimal Transport (OT) Problem). Given two marginal constraints π 0 ,π T ∈P(R d ), the dynamic optimal transport (OT) problem aims to find the optimal probability flowp ⋆ t :R d ×[0,T]→ R ∈ P(R d ) and velocity fieldv ⋆ t :R d ×[0,T]→ R d for t∈ [0,T ] that minimizes: inf (p t ,v t ) Z T 0 Z R d ∥v t (x)∥ 2 p t (x)dxdt s.t. ( ∂ t p t +∇· (p t v t ) = 0 p 0 = π 0 , p T = π T (Dynamic OT Problem) whre the continuity equation∂ t p t +∇·(p t v t ) = 0 enforces conservation of probability mass over the continuous flow. The solution to (Dynamic OT Problem) is a time-interpolation of the optimal transport mapM ⋆ defined as: M ⋆ t (x 0 ) = (1− t)x 0 + tM ⋆ (x 0 ), x 0 ∼ π 0 , t∈ [0,T ](Optimal Transport Map) 28 Figure 3: Comparison Between Static and Dynamic Optimal Transport. Illustration of the relationship between the static and dynamic formulations of optimal transport between two marginal distributionsπ 0 andπ T . The static formulation minimizes the transport cost directly between endpoints via the quadratic cost∥x T −x 0 ∥ 2 , while the dynamic (Benamou–Brenier) formulation instead seeks a time-dependent velocity fieldv t (x) that continuously transports mass fromπ 0 toπ T while minimizing the kinetic energy R T 0 ∥v t (x)∥ 2 dt. The white trajectory represents a particle path under the optimal flow, illustrating how dynamic OT realizes the same optimal coupling as static OT through continuous mass evolution. which yields the optimal marginal densityp ⋆ t as the pushforward ofπ 0 via the transport map at time t∈ [0,T ] denoted p ⋆ t = (M ⋆ t ) # π 0 . This Bernamou-Brenier (dynamic) formulation characterizes optimal transport minimization of kinetic energy over straight, deterministic flows satisfying the continuity equation. However, most real-world systems do not naturally evolve in straight lines, but rather stochastic paths that traverse non-linear manifolds. This naturally leads us to the dynamic Schr ̈odinger bridge problem, which considers the optimal transport problem where the underlying dynamics are stochastic rather than deterministic. 2.2 Dynamic Schr ̈odinger Bridge Problem While the dynamic optimal transport (OT) formulation characterizes the most efficient deterministic flow that transports probability mass between two distributions, many real-world systems evolve under intrinsic stochasticity. This motivates a stochastic generalization of dynamic OT, leading to the dynamic Schr ̈odinger bridge (SB) problem. Instead of searching over deterministic velocity fields, the SB formulation asks: Among all stochastic evolutions that transform an initial distribution into a target distribution over a time horizon [0,T], which one is most likely relative to a given reference dynamics? To answer this, the dynamic SB problem introduces a reference path measureQwhich describes the baseline stochastic dynamics of a system. The dynamic Schr ̈odinger bridge problem then selects, among all processes matching prescribed initial and terminal marginals, the one that deviates minimally from the reference process in relative entropy. Definition 2.2 (Dynamic Schr ̈odinger Bridge Problem). Letπ 0 ,π T ∈P(R d ) be probability measures on state spaceR d and letQ ∈ P(C([0,T];R d )) be a reference path measure, where P(C([0,T ]; R d )) is the space of probability paths over R d . 29 Figure 4: Brownian Motion and Controlled Itˆo Processes. Illustration of one-dimensional stochastic trajectories generated by two stochastic differential equations (SDEs) starting fromX 0 = 0 over the time intervalt∈[0,1]. Left: Sample paths of pure Brownian motion of the formdX t =σ t dB t , where increments are Gaussian with variance proportional to the timestepB t+∆t =B t + √ ∆tzwithz ∼N(0,I d ). Right: Sample paths of a controlled Itˆo process of the formdX t = (f(X t ,t) +σ t u(X t ,t)) +σ t dB t , where we setf ≡0 andu(x,t) := 2−x T−t+ε which pulls the process to X T = 2. The dynamic Schr ̈odinger bridge (SB) problem seeks a new path measureP∈P(C([0,T];R d )) with initial and terminal marginals matchingp 0 =π 0 andp T =π T which minimizes the relative entropy with respect to Q: P ⋆ =arg min P∈P(C([0,T ];R d )) KL(P∥Q) : p 0 = π 0 ,p T = π T (Dynamic SB Problem) where KL(·∥·) denotes the KL divergence on path space given by: KL(P∥Q) = E P log dP dQ (2.2) This perspective reveals the Schr ̈odinger bridge as an entropy-regularized analogue of the (Dynamic OT Problem). In the limit of vanishing noise, the problem recovers classical optimal transport, while for positive noise, it admits a rich stochastic structure. In particular, when the reference dynamics is Brownian motion, the optimal path measure inherits a Markov structure and can be characterized through a time-dependent drift correction of the reference process. In this section, we will break down these ideas, starting with path measures and Itˆo processes, then deriving the path-space KL divergence, and finally exploring the dynamic SB problem through the lens of the Hamilton-Jacobi-Bellman and Fokker-Planck system or equations and the Hopf-Cole transform. 2.3 Path Measures and Itˆo Processes Before diving deeper into Schr ̈odinger Bridge theory, let’s first establish the foundations of path measures and Itˆo processes that will help us solve the SB problem. First, we define a stochastic process as a random variableX t ∈ R d that evolves over a time horizont ∈[0,T], denoted X 0:T := (X t ) t∈[0,T ] . The distribution of many such random variables follows a time-dependent marginal probability distribution, denoted p t ∈P(R d ). Specifically, we want to consider only stochastic processesX 0:T that depend only on the past and present states of the system, which are formally said to be adapted to the filtration (F t ) t∈[0,T ] . The filtration (F t ) t∈[0,T ] is simply a sequence of eventsF t that is ordered such that all events inF s fors≤ tare inF t , i.e.,F s ⊆F t fors≤ t. This definition allows us to define aF t -adapted process which depends only on the past and present events. 30 Definition 2.3 (F t -Adapted Process). A stocastic processX 0:T is said to beF t -adapted if the random variable X t is F t -measurable for all t∈ [0,T ], where: F t := σ(X τ : 0≤ τ ≤ t)(2.3) which is the sigma-algebra a generated by the history of the process up to timetor all the information available from the process up to timet. Equivalently, this means that for all measurable sets S ∈S(R d ) over the state space, the process X 0:T satisfies: X t ∈ S∈F t (2.4) which is a formal way of stating that all valuesX t ∈ Scan be determined using only information up to time t. a This is not to be confused with theσnotation for the diffusion coefficient. We will only useσto denote sigma-algebra sparingly. A specific stochastic process that we will see often is Brownian motion, or F t -adapted Wiener process, which begins at the origin and evolves via independent Gaussian steps with variance that is proportional to the time increment. Definition 2.4 (Brownian Motion). Brownian motion (B t ) t∈[0,T ] is a type of stochastic process that starts atB 0 = 0 and evolves via independent Gaussian increments defined as B t+∆t = B t + √ ∆tz, z ∼N (0,I d )(2.5) wherezis sampled independently from a unit isotropic Gaussian with zero-mean across all time steps. This means that each increment is GaussianB t+∆t − B t ∼ N(0,∆tI d ) and independent of F t . Now that we have defined the notion of stochastic processes and Brownian motion, we can define an Itˆo process which is the class of stochastic processes that are the foundation of most generative modeling frameworks, from diffusion to flow matching to Schr ̈odinger bridge matching. Definition 2.5 (Itˆo Process). An Itˆo Process is a stochastic process (X t ) t∈[0,T ] whose state X t can be written as X t = X 0 + Z t 0 f (X t ,t)ds + Z t 0 Σ t dB t (2.6) which can be equivalently defined as the solution of an stochastic differential equation (SDE) of the form dX t = f (X t ,t)dt + Σ t dB t (2.7) wheref(X t ,t) :R d ×[0,T]→ R d is known as the drift and Σ t ∈ R d×d is the diffusion coefficient matrix. In most applications, the diffusion coefficient is a constant or time-dependent scalar σ t : [0,T ]→ R ≥0 a , which simplifies the SDE into dX t = f (X t ,t)dt + σ t dB t . a While we introduce the theory of path measures with the general covariance matrix Σ t , we simplify to the scalar time-dependent diffusion coefficientσ t in the remainder of this guide as it is most commonly used throughout the literature. 31 Next we consider stochastic dynamics where there exists some external influences that perturb the drift of the process via some control drift. These controlled Itˆo processes can be modeled with a controlled stochastic differential equation (SDE), where the time-dependent control velocity u(x,t) : R d × [0,T ]→ R d enters the drift in the directions spanned by the diffusion coefficient σ t . Definition 2.6 (Controlled Itˆo Process). A controlled Itˆo process is obtained by introducing a controlu(x,t) :R d ×[0,T]→ R da that modifies the drift in the directions spanned by the diffusion coefficient, which yields the controlled stochastic differential equation (SDE) of the form: dX u t = (f (X u t ,t) + Σ t u(X u t ,t))dt + Σ t dB t (Controlled SDE) wheref(x,t) :R d ×[0,T]→ R d is the reference drift and Σ t : [0,T]→ R d×d is the diffusion coefficient. Equivalently, the controlled process can be written as: X u t = X 0 + Z t 0 (f (X u τ ,τ ) + Σ τ u(X u τ ,τ ))dτ + Z t 0 Σ τ dB τ (2.8) a In the remainder of this guide, we omit explicitly writing the domain and codomain (R d ×[0,T]→ R d ) when it is clear from context. Observe that the control is scaled by the diffusion coefficient Σ t urather than added independently. This parameterization of the controlled SDE guarantees that the controlled process remains abso- lutely continuous with respect to the reference path measureQdefined using the diffusion term σ t dB t . This property ensures that the relative entropy used in the dynamic SB objective takes a tractable form, which we will show explicitly in Section 2.5. Intuitively, scaling with the diffusion coefficient ensures that the control drift only steers the process in directions within the space supported by the stochastic noise, whereas adding the control drift independently without scaling could push the process in directions where the noise has no support. Having developed an understanding of uncontrolled and controlled Itˆo processes, we now derive a key property of Itˆo processes, which states that applying some function to the state produces another Itˆo process over the transformed coordinates. This property is captured via Itˆo’s formula. Theorem 2.7 (Itˆo’s Formula inR d ). Consider an Itˆo process (X t ∈ R d ) t∈[0,T ] and a scalar functionφ(x,t) :R d ×[0,T]→ Rthat twice continuously differentiableφ∈ C 2,1 (R d ×[0,T]) a . Then, the transformed random variableφ(X t ,t) is also an Itˆo process that evolves via the SDE given by dφ(X t ,t) = ∂ t φ(X t ,t) +∇φ(X t ,t)dX t + 1 2 dX ⊤ t (∇ 2 φ(X t ,t))dX t (2.9) Substituting dX t = f (X t ,t)dt + Σ t dB t , we rewrite the formula as: dY t = ∂ t φ(X t ,t) +f (X t ,t) ⊤ ∇φ(X t ,t) + 1 2 Tr Σ t Σ ⊤ t ∇ 2 φ(X t ,t) dt +∇φ(X t ,t) ⊤ ΣdB t (Itˆo’s Formula) which means that the space of Itˆo processes is closed under twice-differentiable functions. Ifφ(x) :R d → R∈ C 2 (R d ) is a scalar function dependent only onx, then the partial derivative ∂ t φ disappears, and φ(X t ) follows the SDE: dφ(X t ) = f (X t ,t) ⊤ ∇φ(X t ,t) + σ 2 t 2 ∆φ(X t ) dt +∇φ(X t ) ⊤ σ t dB t (2.10) a functions in the Hessian are continuous and produces a square integrable random variableφ(X t ,t)∈ L 2 (i.e., E[|φ(X t ,t)| 2 ] <∞ 32 Proof. In this proof, we will break down the steps from (2.9) to (Itˆo’s Formula). After substituting dX t = f t dt + Σ t dB t where we denote f t ≡ f (X t ,t) for simplicity, we get: dφ(X t ,t) = ∂ t φ(X t ,t) +∇φ(X t ,t) ⊤ (f t dt + Σ t dB t ) + 1 2 (f t dt + Σ t dB t ) ⊤ (∇ 2 φ(X t ,t))(f t dt + Σ t dB t ) (2.11) Itˆo’s calculus defines dtdt = B t = 0 and (dt) 2 = 0, so we can simplify the second term to: dφ(X t ,t) = ∂ t φ(X t ,t) +∇φ(X t ,t) ⊤ (f t dt + Σ t dB t ) + 1 2 (Σ t dB t ) ⊤ (∇ 2 φ(X t ,t))(Σ t dB t ) | z (⋆) (2.12) Now, we will simplify the (⋆) term in the equation. Recall that Σ t ∈ R d×d ,dB t ∈ R d , and ∇ 2 φ(X t ,t)∈ R d×d by decomposing it into component form. Substituting the matrix-vector product in component form (Σ t dB t ) i = P d k=1 Σ ik t dB k t into (⋆), we have: (Σ t dB t ) ⊤ (∇ 2 φ(X t ,t))(Σ t dB t ) = d X i,j=1 " d X k=1 Σ ik t dB k t ! (∇ 2 φ(X t ,t)) ij d X ℓ=1 Σ jℓ t dB ℓ t !# = X i,j,k,ℓ Σ ik t Σ jℓ t (∇ 2 φ(X t ,t)) ij dB k t dB ℓ t | z δ kℓdt = X i,j,k Σ ik t Σ jk t | z (Σ ⊤ ) ij (∇ 2 φ(X t ,t)) ij dt = X i,j (∇ 2 φ(X t ,t)) ij (Σ ⊤ ) ij | z Tr(Σ ⊤ ∇ 2 φ(X t ,t)) dt = Tr(Σ ⊤ ∇ 2 φ(X t ,t))dt(2.13) Substituting the simplified (⋆) back into (2.12), we get the final form of the SDE for the transformed Itˆo process: dφ(X t ,t) = ∂ t φ(X t ,t) +∇φ(X t ,t) ⊤ (f t dt + Σ t dB t ) + 1 2 Tr Σ t Σ ⊤ t ∇ 2 φ(X t ,t) dt = ∂ t φ(X t ,t) +f (X t ,t) ⊤ ∇φ(X t ,t) + 1 2 Tr Σ t Σ ⊤ t ∇ 2 φ(X t ,t) dt +∇φ(X t ,t) ⊤ Σ t dB t (2.14) which is Itˆo’s formula for an arbitrary twice-continuous functionφ(x,t) :R d ×[0,T]→ R ∈ C 2,1 (R d × [0,T ]) and SDE dX t = f (X t ,t)dt + Σ t dB t . Whenφdoesn’t depend ontand is only a function of the stateφ(x) :R d → R, we have∂ t φ(x) = 0. Additionally, when the diffusion is an isotropic Gaussian Σ t =σ t I d , then the trace term reduces to: Tr Σ t Σ ⊤ t ∇ 2 φ(X t ,t) = Tr σ 2 t I d I ⊤ d ∇ 2 φ(X t ,t) = σ 2 t Tr ∇ 2 φ(X t ,t) = σ 2 t ∆φ(X t ,t)(2.15) With these simplifications, the Itˆo formula becomes: dφ(X t ,t) = f (X t ,t) ⊤ ∇φ(X t ,t) + σ 2 t 2 ∆φ(X t ) dt +∇φ(X t ,t) ⊤ σ t dB t (Simplified Itˆo Process) which is the key ingredient to several derivations and proofs that we will study in this guide. □ Since the Schr ̈odinger bridge problem studies controlled stochastic processes, we establish the following Corollary which is a direct consequence of our proof for (Itˆo’s Formula), but simply replaces the drift of the uncontrolled SDEf(X t ,t) with the drift of the controlled SDE (f(X t ,t) +σ t u(X t ,t)) from Definition 2.6. 33 Corollary 2.8 (Itˆo Formula for Controlled SDEs). Consider a controlled stochastic process (X t ∈ R d ) t∈[0,T ] generated by the controlled SDE: dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB t (Controlled SDE) whereu(x,t) is the control drift andσ t is the scalar diffusion coefficient. For any scalar functionφ(x,t) :R d ×[0,T]→ R ∈ C 2,1 (R d ×[0,T]) a , the transformed stochastic process defined by φ(X u t ,t) follows the SDE: dφ(X u t ,t) = ∂ t φ(X u t ,t) + (f + σu)(X u t ,t) ⊤ ∇φ(X u t ,t) + σ 2 t 2 ∆φ(X u t ,t) dt +∇φ(X u t ,t) ⊤ σ t dB t (Controlled Itˆo Formula) Ifφ(x) :R d → R∈ C 2 (R d ) is a scalar function dependent only onx, then the partial derivative ∂ t φ disappears, and φ(X u t ) follows the SDE: dφ(X u t ) = (f + σu)(X u t ) ⊤ ∇φ(X u t ,t) + σ 2 t 2 ∆φ(X u t ) dt +∇φ(X u t ) ⊤ σ t dB t (2.16) a whereC 1,2 denotes that the function is continuously differentiable with respect totand twice continuously differentiable with respect to x The controlled Itˆo formula shows that, for any sufficiently smooth test function, the drift term of the transformed process is determined by a linear differential operator acting on that function. This operator is called the infinitesimal generator of the SDEA t , which describes the instantaneous rate of change of the expected value of φ(X t ,t) at time t, explicitly written as: (A t φ)(x,t) = lim ∆t→0 E[φ(X t+∆t ,t + ∆t)|X t = x]− φ(x,t) ∆t (Infinitesmal Generator) which plays a central role in describing the local evolution of observables. Definition 2.9 (Infinitesimal Generator of Itˆo Process). For any test functionφ(x,t)∈ C 2,1 (R d ×[0,T]), the infinitesimal generator of the uncontrolled SDE is the operator A t defined as: (A t φ)(x,t) := f (x,t) ⊤ ∇φ(x,t) + σ 2 t 2 ∆φ(x,t)(Uncontrolled Generator) and the infinitesimal generator of the controlled SDE is the operator A u t defined as: (A u t φ)(x,t) := (f (x,t) + σ t u(x,t)) ⊤ ∇φ(x,t) + σ 2 t 2 ∆φ(x,t)(Controlled Generator) Equivalently, the controlled generator can be written in terms of the uncontrolled generator by expanding the divergence as: (A u t φ)(x,t) = f (x,t) ⊤ ∇φ(x,t) + σ 2 t 2 ∆φ(x,t) + σ t u(x,t) ⊤ ∇φ(x,t) = (A t φ)(x,t) +u(x,t) ⊤ ∇φ(x,t)(2.17) This definition of the infinitesimal generator can be used to rewrite Itˆo’s formula for uncontrolled and controlled SDEs as: dφ(X t ,t) = (∂ t +A t )φ(X t ,t)dt + σ t ∇φ(X t ,t) ⊤ dB t (Uncontrolled Itˆo Formula with A t ) dφ(X u t ,t) = (∂ t +A u t )φ(X u t ,t)dt + σ t ∇φ(X u t ,t) ⊤ dB t (Controlled Itˆo Formula with A u t ) 34 Using the infinitesimal generator, we can define a specific family of stochastic processes, called martingales. Definition 2.10 (Martingales). AF t -adapted stochastic processY 0:T := (Y t ) t∈[0,T ] is called a martingale if it is integrable for allt∈[0,T] (i.e.,E[|Y t |]<∞) and satisfies the martingale property: E[Y t |F s ] = Y s =⇒ E[Y t ] = E[Y s ], ∀0≤ s≤ t(Martingale Property) which means that the expectation is constant over time. Furthermore, we have that if Y t :=φ(X t ,t) is the martingale process generated by the functionφ(x,t)∈ C 2,1 (R d ×[0,T]), the uncontrolled generator vanishes, such that: (A t φ)(X t ,t) = f (X t ,t) ⊤ ∇φ(X t ,t) + σ 2 t 2 ∆φ(X t ,t) = 0(2.18) Given the definition of Itˆo processes as stochastic differential equations with drift and diffusion terms, we observe that there are infinitely many possible paths under the same SDE due to the non-deterministic diffusion term, which injects randomness into each trajectory; however, certain paths are more likely than others under a given drift. Sampling an infinite number of paths under a specific SDE forms a probability distribution in the path space, known as a path measureP∈P(C([0,T];R d )), which given an Itˆo processX 0:T ∈ C([0,T];R d ) returns the probability that the process was generated under that SDE. Specifically, we will consider the general space of controlled path measures with a control driftu(x,t), where u≡ 0 yields the reference path measure Q. Definition 2.11 (Controlled Path Measure). A controlled path measureP u ∈ C([0,T];R d ) is a probability measure on the path spaceC([0,T];R d ) induced by a stochastic differential equation (SDE) with controlσ t u(x,t) defined in (Controlled SDE). For any measurable set of trajectoriesS ⊆ C([0,T];R d ),P u (S) is the probability that the trajectories in the set are generated under the controlled SDE: P u (S) = Pr (X 0:T ∈ S under the controlled SDE of P u )(2.19) Throughout this guide, we will use a slight abuse of notation and denote the probability of a stochastic path under P u as P u (X 0:T ). Next, we will define how the marginal densities evolve over the continuous time interval via the Fokker- Planck equation, which provides an alternative representation of path measures as deterministic partial differential equations instead of non-deterministic SDEs. 2.4 Fokker-Planck and Feynman-Kac Equations Defining a path measure with an SDE only captures the evolution of single particles along the trajectories supported by the path measure, but tells us nothing about the evolution of the distribution generated by many particles under a path measure. The Fokker-Planck equation allows us to define the behavior of the probability distribution of particlesp t at each time pointt ∈[0,T], such that individual states simulated by the SDE in (2.7) are samples X t ∼ p t . 35 Theorem 2.12 (Fokker-Planck Equation). LetX 0:T be a stochastic process governed by the SDE: dX t = f (X t ,t)dt + σ t dB t (2.20) wheref(x,t) is the drift andσ t ∈ Ris the scalar diffusion coefficient. Then, the marginal densityp t ∈ P(R d ) of the particles generated by the SDE evolves via the Fokker-Planck equation defined as: ∂ t p t (x) =−∇· (f (x,t)p t (x)) + σ 2 t 2 ∆p t (x)(Fokker-Planck Equation) where X t ∼ p t . Equivalently, p t is probability law of X t , denoted Law(X t ) = p t . Intuition. Before we derive the Fokker-Planck equation explicitly, let us first consider the problem of characterizing the distribution over particles generated from an SDE without direct access to the densityp t . From Itˆo’s formula (Theorem 2.7), we can define the evolution of samples from the SDE via twice-differentiable functions over the state space R d . With the capability to study the random variable with arbitrary smooth functions, known as test functionsφ(x) :R d → R, we can extract different pieces of information about the distributionp t by taking the expectationE p t [φ(X t )]. For instance, the expectation of the identity functionφ(x) =x measures how the mean of the distribution evolves over time, and the squared functionφ(x) =x 2 measures how the variance evolves over time. Unlike the test function used to derive Itˆo’s formula, we don’t require it to depend on time, since we use it only to probe the spatial structure of the density at each time. Intuitively, by analyzing the evolution of any arbitrary smooth functions on X t , we can derive the evolution of the density p t . Derivation. Consider a test functionφ(x)∈ C 2 c (R d ) that is twice differentiable and has compact support such that the boundary terms vanish in integration by parts. By Itˆo’s formula in (Simplified Itˆo Process), the random variable φ(X t ) is an Itˆo process that evolves via the SDE: dφ(X t ) = f (X t ,t) ⊤ ∇φ(X t ) |z drift contribution + σ 2 t 2 ∆φ(X t ) |z diffusion contribution dt + σ t ∇φ(X t ) ⊤ dB t | z stochastic term (2.21) Since we want to define the evolution on a distribution level, we take the expectation on both sides, which eliminates the stochastic term since the Itˆo integral has zero expectation: E p t [dφ(X t )] = E p t f (X t ,t) ⊤ ∇φ(X t ) + σ 2 t 2 ∆φ(X t ) dt + E p t h σ t ∇φ(X t ) ⊤ dB t i | z =0 = E p t f (X t ,t) ⊤ ∇φ(X t ) + σ 2 t 2 ∆φ(X t ) dt(2.22) Dividing both sides by dt, we have: ∂ t E p t [φ(X t )] = E p t f (X t ,t) ⊤ ∇φ(X t ) + σ 2 t 2 ∆φ(X t ) (2.23) By definition, the expectations can be written in integral form asE p t (x) [·] = R R d (·)p t dxwhich gives: ∂ t Z R d φ(x)p t (x)dx = Z R d f (x,t) ⊤ ∇φ(x) + σ 2 t 2 ∆φ(x) p t (x)dx(2.24) Z R d φ(x)∂ t p t (x)dx = Z R d f (x,t) ⊤ ∇φ(x)p t (x)dx | z drift term + 1 2 Z R d σ 2 t ∆φ(x)p t (x)dx | z diffusion term (2.25) 36 For the drift term,f(x,t)∈ R d and∇φ(x)∈ R d , we can apply integration by parts on each coordinate to get: Z R d f (x,t)·∇φ(x)p t (x)dx = Z R d d X i=1 f i (x,t) ∂φ(x) ∂x i p t (x)dx = Z R d d X i=1 ∂φ(x) ∂x i |z dv f i (x,t)p t (x) |z u dx(1st integration by parts) =− Z R d d X i=1 φ(x) | z v ∂ ∂x i f i (x,t)p t (x) | z du dx(2nd integration by parts) =− Z R d φ(x) d X i=1 ∂ ∂x i f i (x,t)p t (x) | z =∇·(f (x,t)p t (x)) dx = − Z R d φ(x)∇· (f (x,t)p t (x))dx(drift term) where all boundary terms vanish sinceφhas compact support. For the diffusion term, we apply integration by parts twice as follows: 1 2 Z R d σ 2 t ∆φ(x)p t (x)dx = σ 2 t 2 Z R d p t (x) d X i=1 ∂ 2 ∂x i ∂x i φ(x)dx = σ 2 t 2 d X i=1 Z R d p t (x) | z u ∂ ∂x i ∂ ∂x i φ(x) |z dv dx(1st integration by parts) =− σ 2 t 2 d X i=1 Z R d ∂ ∂x i p t (x) | z u ∂ ∂x i φ(x) | z dv dx(2nd integration by parts) = σ 2 t 2 d X i=1 Z R d φ(x) ∂ 2 ∂x i ∂x i p t (x)dx = σ 2 t 2 Z R d φ(x) d X i=1 ∂ 2 ∂x i ∂x i p t (x) |z ∆p t (x) dx = σ 2 t 2 Z R d φ(x)∆p t dx (diffusion term) Substituting the simplified drift and diffusion terms back into (2.25), we obtain: ∂ t Z R d φ(x)p t (x)dx =− Z R d φ(x)∇· (f (x,t)p t (x))dx |z drift term + 1 2 Z R d φ(x)σ 2 t ∆p t (x)dx |z diffusion term = Z R d φ(x) −∇· (f (x,t)p t (x)) + σ 2 t 2 ∆p t (x) dx(2.26) Since we definedφ(x) such that this equality holds for any arbitrary test function, then the integrands must be equal which recovers the Fokker-Planck (FP) equation: ∂ t p t (x) =−∇· (f (x,t)p t (x)) + σ 2 t 2 ∆p t (x)(Fokker-Planck Equation) 37 Furthermore, the densityp t that solves the FP equation is unique givne a fixed initial condition 9 .□ By testing against an arbitrary smooth test function and equating integrands, we recover the Fokker–Planck equation that governs the time evolution of the densityp t (x). We observe that the drift termf(x,t) induces the transport of mass via divergence, while the diffusion coefficientσ t contributes to smoothing through the Laplacian operator. The Fokker–Planck equation therefore provides a forward-time description of the stochastic dynamics, complementing the underlying stochastic differential equation. Remark 2.13 (Fokker-Planck Equation Generalizes the Continuity Equation). The Fokker- Planck equation can be interpreted as a stochastic generalization of the classic continuity equation from the (Dynamic OT Problem), which ensures the conservation of probability mass, defined as: ∂ t p t (x) +∇· p t (x)v(x,t) = 0(Continuity Equation) which expresses that probability mass is transported along the flow induced by the velocity v(x,t). The Fokker-Planck equation is equal to the continuity equation with an additional Laplacian term σ 2 t 2 ∆p t that captures the spreading of mass caused by Brownian motion. Intuitively, the Laplacian is the divergence of the gradient ∆p t =∇·∇p t which measures whether the direction of steepest increase in probability density∇p t is spreading out (positive divergence) or converging (negative divergence), so the positive diffusion coefficient increases the spreading out of probability density. Using this intuition, we can easily extend the Fokker-Planck equation to controlled processes by introducing the scaled controlσ t u(x,t) into the drift contribution, yielding the controlled Fokker- Planck equation that governs the evolution of the controlled marginal density. Corollary 2.14 (Controlled Fokker-Planck Equation). Let (X u t ) t∈[0,T ] be a stochastic process governed by the SDE: dX u t = (f (X u t ,t) + σ t u(X u t ,t))dt + σ t dB t ,(2.27) whereu(x,t) is the control drift,f(x,t) is the reference drift andσ t ∈ Ris the scalar diffusion coefficient. Letp t ∈P(R d ) denote the marginal density or probability law ofX u t , such that X u t ∼ p t andLaw(X u t ) =p t . Thenp t evolves according to the controlled Fokker-Planck equation ∂ t p t (x) =−∇· ((f (x,t) + σ t u(x,t))p t (x)) + σ 2 t 2 ∆p t (x)(Controlled Fokker-Planck Equation) Equivalently, expanding the divergence term gives ∂ t p t (x) =−∇· (f (x,t)p t (x))−∇· σ t u(x,t)p t (x) + σ 2 t 2 ∆p t (x)(2.28) We have shown that uncontrolled and controlled Itˆo processes can not only be written as stochastic differential equations (SDEs) that govern the forward-time evolution of individual trajectoriesX t , but also admit an equivalent deterministic representation through the Fokker-Planck equation, which is a deterministic partial differential equation (PDE) that governs the forward-time evolution of the marginal density p t that characterizes the probability law of the SDE trajectories. 9 For rigorous proof of uniqueness, see Bogachev et al. (2021) 38 However, the Schr ̈odinger bridge problem is inherently a two-sided problem enforced by both initial and terminal marginal distribution constraints. Therefore, we need to consider how constraints or functions evaluated on the terminal distributionp T evolve backward in time through the stochastic process to determine the probability of an intermediate state under the terminal constraint. This evolution is captured by the Feynman-Kac equation, which we analyze next. Theorem 2.15 (Feynman-Kac Equation). Consider a scalar functionr(x,t) :R d ×[0,T]→ R that solves the linear PDE with a terminal constraintr(x,T) = Φ(x) and running cost c(x,t) : R d × [0,T ]→ R a defined as: ∂ t r(x,t) +⟨f (x,t),∇r(x,t)⟩ + σ 2 t 2 ∆r(x,t)− c(x,t)r(x,t) = 0, r(x,T ) = Φ(x) (Feynman-Kac PDE) wheref(x,t) is the reference drift andσ t is the diffusion coefficient of an SDE that generates a stochastic process (X τ ) τ∈[t,T ] defined by: dX τ = f (X τ ,τ )dτ + σ τ dB τ , X t = x(2.29) Then, the solutionrthat solves the (Feynman-Kac PDE) can be written as the following expectation over stochastic paths: r(x,t) = E " exp − Z T t c(X s ,s)ds ! Φ(X T ) X t = x # (Feynman-Kac Formula) which is called the Feynman-Kac formula. a Note that from now on,c(x,t) will be used to denote the running cost, which is distinct fromc(x,y) which is used in Section 1 to denote the transport cost from x to y Derivation. Intuitively,r(x,t) is the expected future value of the terminal constraint Φ(X T ) after discounting the running costc(X s ,s) given the current stateX t =xat timet. With this intuition in mind, we derive the Feynman-Kac formula in steps. Step 1: Define the Stochastic Processes. We can consider the following stochastic process that describes the evolution of the accumulated running cost over [τ,t]: M τ := exp − Z τ t c(X s ,s)ds |z =:Z τ r(X τ ,τ ), τ ∈ [t,T ](2.30) Since the exponential term is itself a stochastic process that changes withτ, we can defineZ τ := exp − R τ t c(X s ,s)ds andM τ :=Z τ r(X τ ,τ). Now, we aim to show thatM τ is a martingale whose expected value does not change over time (Definition 2.10). In other words, we aim to show that the expected future reward E[M T ] is equal to the current value M t , i.e., E[M T ] = M t . Step 2: Apply Itˆo’s Formula. Applying Itˆo’s product rule toM s andr(X s ,s), we get the following expression for dM s : dM τ = Z τ dr(X τ ,τ ) + r(X τ ,τ )dZ s + dZ s dr(X τ ,τ ) |z =O(dτ 2 )=0 = Z τ dr(X τ ,τ ) |z (⋆) +r(X τ ,τ )dZ τ | z (⋄) (2.31) 39 To expand (⋆), we can apply (Itˆo’s Formula) to get: dr(X τ ,τ ) = ∂ τ r(X τ ,τ ) +⟨f (X τ ,τ ),∇r(X τ ,τ )⟩ + σ 2 t 2 ∆r(X τ ,τ ) |z uncontrolled generator applied to r dτ +∇r(X τ ,τ ) ⊤ σ t dB τ = (∂ τ r(X τ ,τ ) +A t r(X τ ,τ ))dτ +∇r(X τ ,τ ) ⊤ σ t dB τ (⋆) where we recognize the unconditional generatorA t from (Uncontrolled Generator). To expand (⋄), we denote A τ :=− R τ t c(X s ,s)ds and apply the chain rule: dZ τ = d(e A τ ) = e A τ |z =Z τ dA τ = Z τ dA τ dτ dτ = Z τ (−c(X τ ,τ )dτ ) =−c(X τ ,τ )Z τ dτ(⋄) where the expression for dA τ dτ =−c(X τ ,τ) follows from the fundamental theorem of calculus for integrals with variable upper limit. Substituting (⋆) and (⋄) back into (2.31), we get: dM τ = Z τ h (∂ τ r +Ar)dτ +∇r ⊤ σ t dB τ i | z (⋆) +r (−cZ τ dτ ) | z (⋄) = Z τ (∂ τ r +Ar− cr) |z =0 dτ +Z τ ∇r ⊤ σ t dB τ (2.32) where we observe that the drift term is exactly equal to the (Feynman-Kac PDE), which is equal to zero given the solutionr(x,t). Therefore, the drift disappears, anddM τ =Z τ ∇r ⊤ σ t dB τ contains only a diffusion term and is a martingale with constant zero-expectation. Therefore, we have: M T = M t + Z T t dM τ = M t + Z T t Z τ ∇r ⊤ σ t dB τ =⇒ E[M T ] = E[M t ] + E Z T t Z τ ∇r ⊤ σ t dB τ |z Itˆo integral =0 =⇒ E[M T ] = E[M t ] = M t (2.33) where the final equality follows from the fact thatM t is the starting time, and the accumulated cost over [t,t] is 0. Step 3: Substitute Terminal Constraint. By our definition in (2.30), we have: M T = exp − Z T t c(X s ,s)ds r(X T ,T ) |z =Φ(X T ) = exp − Z T t c(X s ,s)ds Φ(X T )(2.34) Given thatM t =r(x,t) andM t =E[M T ], we have shown that the solutionr(x,t) takes the form: r(x,t) = E exp − Z T t c(X s ,s)ds Φ(X T ) X t = x (Feynman-Kac Formula) which is exactly the (Feynman-Kac Formula).□ A special case of the Feynman-Kac formula when there is no running cost (c≡0) is the Kolmogorov backward equation, which describes the expected future reward as depending only on the expected value over terminal states. Corollary 2.16 (Kolmogorov Backward Equation). Let Φ :R d → Rbe a terminal constraint function and definer(x,t) :R d ×[0,T]→ R d as the expected future reward from a intermediate 40 state X t = x: r(x,t) := E X t:T ∼Q [Φ(X T )|X t = x](Feynman-Kac Representation) whereQis the reference path measure defined by the SDEdX t =f(X t ,t)ds+σ t dB t . Then r(x,t) satisfies the Kolmogorov backward equation defined as: ∂ t r(x,t) +⟨f (x,t),∇r(x,t)⟩ + σ 2 t 2 ∆r(x,t) = 0, r(x,T ) = Φ(x) (Kolmogorov Backward Equation) Together, the (Fokker-Planck Equation) and (Feynman-Kac Formula) describe the two complementary ways in which stochastic differential equations evolve in time. The Fokker–Planck equation characterizes the forward evolution of probability densitiesp t , describing how the distribution of particles transported by an SDE spreads and flows through state space. In contrast, the Feynman-Kac formula provides a backward evolution of functions along stochastic trajectories, expressing solutions to certain partial differential equations as expectations over future paths of the stochastic process. In this sense, the Fokker-Planck equation propagates probability mass forward in time, while the Feynman-Kac equation propagates value functions or potentials backward through the stochastic dynamics. This forward–backward structure will play a central role in the theory of Schr ̈odinger bridges, where the optimal dynamics are characterized by the interaction between forward-evolving densities and backward-evolving potentials that together determine the controlled stochastic process connecting the prescribed endpoint distributions. 2.5 Girsanov’s Theorem Now that we have established path measures and how they evolve both on an individual trajectory level via stochastic differential equations and on a density level via the Fokker-Planck partial differential equation, we will shift our focus to describing the relationship between path measures. This is the core objective of the Schr ̈odinger bridge problem, which is to define a new path measure that minimally diverges from a reference path measure with marginal constraints. We begin our discussion with Girsanov’s theorem, which characterizes the probability density ratio of a single stochastic process under different path measures. In this section, we derive Girsanov’s theorem in the special case where the reference path measure is standard Brownian motion with identity diffusion matrix. The key idea is to understand how a change of drift modifies the underlying path measure. To understand the intuition, we build the result from scratch using the discrete-time structure of Brownian motion 10 . For notational simplicity, we will consider a constant diffusion coefficient σ over time, and omit the time subscript. Recall from Definition 2.4 that Brownian motion can be written as a sequence of independent Gaussian increments which can be written in discrete time with step size ∆t := t k+1 − t k as: σB : B t k+1 = B t k + σ √ ∆tz =: B t k + σ∆B t k , z ∼N (0,I d )(2.35) whereσis the diffusion coefficient andσBto denotes the pure Brownian motion process with diffusion coefficientσ. Then, we can write the Brownian increment ∆B t k as being sampled from ad-dimensional Gaussian with covariance ∆tI d , denoted ∆B t k ∼ N(0,∆tI d ). The transition probability underσBis a Markov transition densityp(B t+∆t |B 0:t ) =p(B t+∆t |B t ) which is a 10 This follows closely from the derivation presented in S ̈arkk ̈a et al. (2019). 41 d-dimensional Gaussian centered at B t k with covariance σ 2 ∆tI d given by: σB(B t k+1 |B t k ) =N (B t k+1 |B t k ) = 1 (2πσ 2 ∆t) d/2 exp − ∥B t k+1 −B t k ∥ 2 2σ 2 ∆t (2.36) Then, we can write the joint probability of the discrete-time Brownian process (B t k ) k∈1,...,K = B t 1 ,...,B t n where t k+1 = t k + ∆t as: σB(B t 1 ,...,B t K ) = K Y k=1 σB(B t k+1 |B t k ) = K Y k=1 " 1 (2πσ 2 ∆t) d/2 exp − ∥B t k+1 −B t k ∥ 2 2σ 2 ∆t !# = K Y k=1 1 (2πσ 2 ∆t) d/2 exp − σ 2 ∥∆B t k ∥ 2 2σ 2 ∆t = K Y k=1 1 (2πσ 2 ∆t) d/2 exp − ∥∆B t k ∥ 2 2∆t (2.37) Now, we consider the Itˆo process with a control driftu(X t ,t) that evolves via the SDE given by (2.7) which evolves in discrete increments defined by: P : X t k+1 = X t k + σu(X t k ,t k )∆t + σ∆B t k (2.38) which is the discrete Brownian process in (2.35) with an extra drift term that generates the path measureP. Since the drift depends only on the current stateX t , the transition probability is also a Markov transition densityP(X t k+1 |(X t ℓ ) ℓ∈1,...,k ) =P(X t k+1 |X t k ) and can be written as a Gaussian centered at X t k + σu(X t k ,t k )∆t with covariance σ 2 ∆tI d given by: P(X t k+1 |X t k ) =N (X t k+1 |X t k + σu(X t k ,t k )∆t) = 1 (2πσ 2 ∆t) d/2 exp − ∥X t k+1 − (X t k + σu(X t k ,t k )∆t)∥ 2 2σ 2 ∆t (2.39) with joint probability density of the full discrete process (X t k ) k∈1,...,K =X t 1 ,...,X t K given by: P(X t 1 ,...,X t K ) = K Y k=1 P(X t k+1 |X t k ) = K Y k=1 " 1 (2πσ 2 ∆t) d/2 exp − ∥X t k+1 − (X t k + σu(X t k ,t k )∆t)∥ 2 2σ 2 ∆t !# = K Y k=1 " 1 (2πσ 2 ∆t) d/2 exp − ∥X t k+1 −X t k − σu(X t k ,t k )∆t∥ 2 2σ 2 ∆t !# (2.40) Now, suppose we have a path (X t k ) k∈1,...,K generated under the Brownian processσBsuch that X t k+1 = X t k + σ∆B t k . Then, we can define the ratio of the path probability under P with respect to σB as: dP dσB (X t k ) k∈1,...,K = P(X t 1 ,...,X t K ) σB(X t 1 ,...,X t K ) = Q K k=1 1 (2πσ 2 ∆t) d/2 exp − ∥X t k+1 −X t k −σu(X t k ,t k )∆t∥ 2 2σ 2 ∆t Q K k=1 h 1 (2πσ 2 ∆t) d/2 exp − ∥∆B t k ∥ 2 2∆t i = K Y k=1 exp − ∥σ∆B t k − σu(X t k ,t k )∆t∥ 2 2σ 2 ∆t + ∥∆B t k ∥ 2 2∆t = K Y k=1 exp − ∥∆B t k −u(X t k ,t k )∆t∥ 2 2∆t + ∥∆B t k ∥ 2 2∆t (2.41) 42 where the second-last equality follows from the fact that the incrementX t k 7→ X t k+1 was made under the reference process such thatX t k+1 =X t k +σ∆B t k , which allowed us to factor out theσ 2 and cancel with the denominator. Expanding∥∆B t k − u(X t k ,t k )∆t∥ 2 =∥∆B t k ∥ 2 −2u(X t k ,t k ) ⊤ ∆B t k ∆t+ ∥u(X t k ,t k )∥ 2 ∆t 2 gives us: dP dσB (X t k ) k∈1,...,K = K Y k=1 exp −∥∆B t k ∥ 2 + 2u(X t k ,t k ) ⊤ ∆B t k ∆t−∥u(X t k ,t k )∥ 2 ∆t 2 +∥∆B t k ∥ 2 2σ 2 ∆t = exp K X k=1 2u(X t k ,t k ) ⊤ ∆B t k ∆t−∥u(X t k ,t k )∥ 2 ∆t 2 2∆t ! = exp K X k=1 u(X t k ,t k )∆B t k − 1 2 K X k=1 ∥u(X t k ,t k )∥ 2 ∆t ! (2.42) which depends only on the Brownian increments ∆B t k and the driftu(X t k ,t k ). Taking the continuous time limit as ∆t→0, we have P k u t k ∆B t k → R T 0 u t dB t and P k ∥u t k ∥ 2 ∆t→ R T 0 ∥u t ∥ 2 dt, resulting in the following form of ratio between path measures: dP dσB (X 0:T ) = exp − 1 2 Z T 0 ∥u(X t ,t)∥ 2 dt + Z T 0 u(X t ,t)dB t ! (2.43) This ratio dP dσB (X 0:T ) is also the Radon-Nikodym derivative (RND) betweenPandσB, which can be used to transform the likelihood of a path underσBto its likelihood underP. From this intuition, we can state the formal version of Girsanov’s theorem (Girsanov, 1960; Karatzas et al., 2014; Øksendal, 2003). Theorem 2.17 (Girsanov’s Theorem). Consider thed-dimensional Brownian motion (B t ) t∈[0,T ] adapted to the filtration (F t ) t∈[0,T ] . Given two path measuresP, P ′ ∈P(R d ), whereP ′ ≪ P, define the density process (Z t ) t∈[0,T ] of the ratio: Z t := E P dP ′ dP F t (2.44) which is the likelihood ratio given the information up to timet. Then, there exists a predictable process (θ s ) s∈[0,T ] such that: Z t = exp − 1 2 Z t 0 ∥θ s ∥ 2 ds + Z t 0 θ ⊤ s dB s , t∈ [0,T ](2.45) Furthermore, the stochastic process (B ′ t ) t∈[0,T ] defined as: B ′ t := B t − Z t 0 θ s ds(Girsanov’s Theorem) is a standard Brownian motion underP ′ , such that for any discrete time intervals 0 =t 0 ≤ · ≤ t k ≤ · ≤ t K ≤ Tthe increments ∆B ′ t k :=B ′ t k+1 − B ′ t k ∼ N (0,(t k+1 − t k )I d ) and independent. Intuitively, Girsanov’s theorem provides a way to change the probability measure on path space while leaving the underlying trajectories unchanged. By definingθ t such that the drift of the original path probability cancels out after applying the transformation, Girsanov’s theorem states that the θ t -tilted process becomes a standard Brownian motion under the transformed probability measure. To ensure that the resulting path measure obtained from reweighting with Girsanov’s theorem is 43 valid, we establish Novikov’s condition, which states an integrability condition that ensures that the exponential density process is a true martingale satisfying (Martingale Property) that has constant expectation and thus defines a valid change of measure. Lemma 2.18 (Novikov’s Condition). Let (B t ) t∈[0,T ] be ad-dimensional Brownian motion adapted to a filtration (F t ) t∈[0,T ] and let (θ t ) t∈[0,T ] be aF t -adapted process that is square integrable over every finite time interval, such that R T 0 ∥θ t ∥ 2 dt <∞ almost surely. Then, if the exponential process given by: Z t = exp − 1 2 Z t 0 ∥θ s ∥ 2 ds + Z t 0 θ ⊤ s dB s (2.46) satisfies the Novikov’s condition: E exp 1 2 Z t 0 ∥θ s ∥ 2 ds <∞(Novikov’s condition) Then, (Z t ) t∈[0,T ] is a true martingale satisfying (Martingale Property) with constant expecta- tion E[Z t ] = 1 for t∈ [0,T ]. The purpose of Novikov’s Condition is to ensureZ t does not lose mass and is properly normalized. In other words, it ensures that the exponential tilt used in Girsanov’s theorem can serve as a valid likelihood ratio between path measures and defines a proper change of probability measure on path space. Leveraging Girsanov’s theorem and Novikov’s Condition, we can now define changes in measure and KL divergences in path space as local drift changes in their stochastic differential equations. 2.6 Path Measure Radon-Nikodym Derivative and KL Divergence To understand the SB problem, it is crucial to understand the measurement used to quantify distance between two path measuresQ, P∈P(C([0,T];R d )) known as the Kullback-Leibler (KL) divergence (also referred to as the relative entropy or I-divergence), denotedKL(P∥Q) (Csisz ́ar, 1975). Definition 2.19 (KL Divergence Between Path Measures). Given two probability measures over the space of paths P, P ′ ∈P(C([0,T ]; R d )), the KL divergence is given by KL(P ′ ∥P) = E P ′ log dP ′ dP = Z C([0,T ];R d ) log dP ′ dP (X 0:T )dP ′ (X 0:T )(2.47) whereX 0:T is a stochastic process and dP ′ dP is the Radon-Nikodym derivative betweenP ′ and P. Since we have defined path measures as solutions to SDEs in Section 2.3, we can now introduce how to write the KL divergence in integral form. First, we derive the Radon-Nikodym (RN) derivative with respect to the drift and diffusion of two Itˆo processes. Theorem 2.20 (Radon-Nikodym Derivative of Controlled Itˆo Path Measures (S ̈arkk ̈a et al., 2019)). Consider two controlled Itˆo path measuresP u andP ̃u with control driftsuand ̃ u , where P ̃u ≪ P ua . Assuming the same reference drift f (x,t) and diffusion coefficient σ t , they 44 Figure 5: Change of Measure Using Radon-Nikodym Derivative. Sample trajectories of a diffusion process are shown under two probability measures. Top left: trajectories under the reference measureQ, governed by the uncontrolled reference driftf(X t ,t). Bottom left: trajectories under the controlled measureP u , where the dynamics include an additional control driftu(X t ,t). Right: the same reference trajectories are reweighted according to the Radon–Nikodym derivative dP u dQ , which assigns higher likelihood to paths aligned with the control and lower likelihood to paths that oppose it. The color intensity represents the log-likelihood ratiolog dP u dQ , illustrating how the controlled dynamics can be interpreted as a change of measure on the same underlying path space. are defined by the SDEs: P u : dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB u t , X 0 = x(u-SDE) P ̃u : d ̃ X t = (f (X t ,t) + σ t ̃ u( ̃ X t ,t))dt + σ t dB ̃u t , ̃ X 0 = x( ̃ u-SDE) The Radon-Nikodym derivative of the corresponding path measures P u and P ̃u is given by dP ̃u dP u (X u 0:T ) = exp − 1 2 Z T 0 ∥( ̃ u−u)(X u t ,t)∥ 2 dt + Z T 0 ̃ u−u (X u t ,t) ⊤ dB u t (Path RND) whereX u 0:T denotes a stochastic process generated underP u andB u t denotes the Brownian motion under P u . a P ̃u ≪ P u means thatP ̃u is absolutely continuous with respect toP u and any event with zero probability under P u also has zero probability under P ̃u Proof. The key idea of the Radon-Nikodym derivative is to define how likely a stochastic process X u 0:T := (X u t ) t∈[0,T ] defined as a standard Brownian motion underP u is under the path measureP ̃u . Since a Brownian motion underP u andP ̃u differ only in their control drifts, we define the change in the drifts as: θ t := ( ̃ u−u)(X u t ,t)∈ R d (2.48) Following (Girsanov’s Theorem), we can define an density process Z t as: Z t := exp − 1 2 Z t 0 ∥θ s ∥ 2 ds + Z t 0 θ ⊤ s dB s s.t. E P u [Z T ] = 1(2.49) 45 where the conditionE P u [Z T ]= 1 follows from the assumption thatZ t satisfies (Novikov’s condition) and is a true martingale. From (Girsanov’s Theorem), we know thatZ T defines the density ratio of a new measure P ′ with respect to P u given by: Z t := E P u dP ′ dP u F t (2.50) Then, according to (Girsanov’s Theorem), we have that the Brownian motionB ′ under the new measure P ′ is a standard Brownian motion defined by: B ′ t = B u t − Z t 0 θ s ds =⇒ dB u t = dB ′ t +θ t dt(2.51) Now, we can rewrite (u-SDE) under the Brownian motion B ′ as: dX u t = (f (X u t ,t) + σ t u(X u t ,t))dt + σ t dB u t = (f (X u t ,t) + σ t u(X u t ,t))dt + σ t (dB ′ t +θ t dt) = (f (X u t ,t) + σ t (u(X u t ,t)+θ t )dt + σ t dB ′ t (2.52) Substituting the definition θ t := ( ̃ u−u)(X u t ,t), we have: dX u t = (f (X u t ,t) + Σ t (u(X u t ,t)+ ̃ u(X u t ,t)−u(X u t ,t))dt + Σ t dB ̃u t = (f (X u t ,t) + Σ t ̃ u(X u t ,t))dt + Σ t dB ̃u t (2.53) which exactly matches ( ̃ u -SDE), so we conclude thatP ′ =P ̃u andB ′ =B ̃u . Then, we have that dP ̃u dP u is equal to Z T given by: dP ̃u dP u (X u 0:T ) = Z T = exp − 1 2 Z t 0 ∥θ s ∥ 2 ds + Z t 0 θ ⊤ s dB u s = exp − 1 2 Z t 0 ∥( ̃ u−u)(X u t ,t)∥ 2 ds + Z t 0 ( ̃ u−u)(X u t ,t) ⊤ dB u s (2.54) which is exactly the Radon-Nikodym derivative that reweights stochastic processes underP u generated with (u-SDE) with their probabilities under P ̃u defined by ( ̃ u-SDE).□ From Definition 2.19, we defined the KL divergence between path measures as the expectation of the logarithm of the Radon-Nikodym derivative between two Itˆo processes. Using Theorem 2.20, we can define the corresponding integral form for the KL divergence. Corollary 2.21 (KL Divergence of Itˆo Path Measures). Consider two controlled path measures P ̃u andP u , whereP ̃u ≪ P u , with control driftsuand ̃ uand SDEs (u-SDE) and ( ̃ u-SDE). The KL divergence of P ̃u with respect to P u is given by: KL(P ̃u ∥P u ) = E X ̃u 0:T ∼P ̃u log dP ̃u dP u (X ̃u 0:T ) = E X ̃u 0:T ∼P ̃u 1 2 Z t 0 ∥ ̃ u(X ̃u t ,t)−u(X ̃u t ,t)∥ 2 ds (Path KL Divergence) whereX ̃u 0:T denotes stochastic trajectories sampled under the law ofP ̃u . WhenP u :=Qis the reference path measure with zero controlu≡0 with SDEdX t =f(X t ,t)dt+σ t dB t , then the KL divergence reduces to: KL(P ̃u ∥Q) = E X ̃u 0:T ∼P ̃u log dP ̃u dQ (X ̃u 0:T ) = E X ̃u 0:T ∼P ̃u 1 2 Z t 0 ∥ ̃ u(X ̃u t ,t)∥ 2 ds (2.55) 46 Proof. Starting from our definition of (Path RND), we can expand the expression for path measure KL divergence from Definition 2.19 as: KL(P ̃u ∥P u ) = E X ̃u 0:T ∼P ̃u log dP ̃u dP u (X ̃u 0:T ) = E X ̃u 0:T ∼P ̃u − 1 2 Z t 0 ∥( ̃ u−u)(X ̃u t ,t)∥ 2 ds + Z t 0 ( ̃ u−u)(X ̃u t ,t) ⊤ dB u s (2.56) where the superscript inX ̃u 0:T is used to explicitly denote that they are stochastic processes generated underP ̃u . Recall that the Itˆo integral of a standard Brownian motion vanishes under the expectation over its associated path measure. However, since the expectation in (2.56) is underP ̃u and the Brownian motion is underP u , we need to rewritedB u s in terms ofdB ̃u s for the Itˆo integral to vanish. To do this, we apply (Girsanov’s Theorem) which states that the Brownian motionB ̃u under the measure P ̃u is a standard Brownian motion defined by: B ̃u t = B u t − Z t 0 θ s ds =⇒ dB u t = dB ̃u t +θ t dt(2.57) where we defined (θ t ) t∈[0,T ] as in Theorem 2.20 as the difference in control driftsθ t := ( ̃ u−u)(X ̃u t ,t ) evaluated along controlled trajectories generated under P ̃u . Therefore, we can rewrite (2.56) as: KL(P ̃u ∥P u ) = E X ̃u 0:T ∼P ̃u − 1 2 Z t 0 ∥( ̃ u−u)(X ̃u t ,t)∥ 2 ds + Z t 0 ( ̃ u−u)(X ̃u t ,t) ⊤ (dB ̃u s + ( ̃ u−u)(X ̃u s ,s)ds) = E X ̃u 0:T ∼P ̃u − 1 2 Z t 0 ∥( ̃ u−u)(X ̃u t ,t)∥ 2 ds + Z t 0 ( ̃ u−u)(X ̃u t ,t) ⊤ dB ̃u s |z vanishes under expectation + Z t 0 ∥( ̃ u−u)(X ̃u t ,t)∥ 2 ds = E X ̃u 0:T ∼P ̃u 1 2 Z t 0 ∥( ̃ u−u)(X ̃u t ,t)∥ 2 ds (2.58) which is the integral form of the path measure KL divergence. By simply substitutingu≡0, we get the KL divergence for the case where P u = Q is the reference path measure no control drift: KL(P ̃u ∥Q) = E X ̃u 0:T ∼P ̃u log dP ̃u dQ (X ̃u 0:T ) = E X ̃u 0:T ∼P ̃u 1 2 Z t 0 ∥ ̃ u(X ̃u t ,t)∥ 2 ds (2.59) which is the form of the KL divergence that we will use to define the dynamic SB objective. □ The Radon–Nikodym derivative between controlled diffusion path measures provides an explicit representation of how changes in drift modify the probability of trajectories (Figure 5). Using (Girsanov’s Theorem), this change of measure produces a quadratic control cost whose expectation reduces to the path-space KL divergence between the two measures. The final integral shows that minimizing the KL divergence between controlled path measures is equivalent to the expectation of the squared difference between the drifts over sampled stochastic paths. 2.7 Schr ̈odinger Bridge with Arbitrary Reference Dynamics Recall from Section 2.3 the family of controlled path measuresP u ∈P(C([0,T];R d )) which characterize the probability law of stochastic processes X 0:T ∈ C([0,T ]; R d ) generated by the controlled SDE: dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB t (Controlled SDE) wheref(x,t) is the reference drift,u(x,t) is the control drift andσ t is the scalar diffusion coefficient. We also defined the reference path measure to be the special case whereu ≡0 and the drift consists only of f (x,t). 47 Within the family of controlled path measures, the dynamic Schr ̈odinger bridge (SB) problem seeks the control driftu ⋆ that minimally perturbs the reference path measure while steering the process between prescribed marginal distributions. More precisely, minimizing the perturbation from the reference path measure is defined as minimizing the KL divergence, which, following Corollary 2.21, is equivalent to minimizing the kinetic energy produced from perturbing the reference dynamics with the control drift. Consequently, the dynamic SB can be interpreted as selecting the control drift that minimally perturbs the reference dynamics while satisfying the marginal distribution constraints X 0 ∼ π 0 and X T ∼ π T . Definition 2.22 (Dynamic Schr ̈odinger Bridge with Arbitrary Reference Dynamics). Let π 0 ,π T ∈ P(R d ) denote the prescribed initial and terminal distributions. The dynamic Schr ̈odinger bridge problem, where the reference dynamics are defined by a driftf, aims to determine the optimal control driftu ⋆ that minimizes the KL divergence of the induced path measures subject to the marginal constraints: inf P u ∈P(C([0,T ];R d )) KL(P u ∥Q) = inf u∈U E X u 0:T ∼P u " Z T 0 1 2 ∥u(X u t ,t)∥ 2 dt # (Dynamic SB) s.t. ( dX u t = (f (x,t) + σ t u(x,t))dt + σ t dB t X u 0 ∼ π 0 , X u T ∼ π T where the KL divergence follows from Corollary 2.21 and we define: (i) U =u ∈ C 1 (R d ×[0,T];R d )|∃C >0, ∀(x,t)∈ R d ×[0,T], u(x,t)≤ C(1 +∥x∥)is the set of all feasible control drifts which contains all smooth control fields that don’t grow faster than linearly, ensuring the controlled stochastic dynamics remain well-behaved (N ̈usken et al., 2021) a . (i) P u ∈P(C([0,T];R d )) as the path measure induced by the controlled SDE andX u 0:T as a stochastic process generated under the controlled SDE with probability law P u . Solving (Dynamic SB) yields the optimal pair (u ⋆ , P ⋆ ) defines the Schr ̈odinger bridge between π 0 and π T relative to f . a For the remainder of this guide, we assume optimization is over the set of feasible controls and omit stating it explicitly as a constraint Rather than an expectation over stochastic processes underP u , we can also write the problem in terms of the time-dependent marginal densitiesp t ∈ P(R d ) defined over the space of probability distributions over the state R d : inf (u,p t ) Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x)dxdt(Density Dynamic SB Problem) s.t. ( ∂ t p t (x) =−∇· p t (x)(f (x,t) + σ t u(x,t)) + σ 2 t 2 ∆p t (x) p 0 = π 0 , p T = π T where the marginal distributions satisfyp 0 =π 0 andp T =π T and the evolution ofp t satisfies the (Controlled Fokker-Planck Equation) derived in Section 2.4. Since the constrained optimization problem in (Density Dynamic SB Problem) requires minimizing with respect to bothu(x,t) andp t (x) which are coupled via the Fokker-Planck PDE constraint, we cannot directly minimize this objective as any variation inu(x,t) will implicitly changep t (x). Therefore, we introduce a Lagrange multiplierψ t that turns (Density Dynamic SB Problem) to 48 an unconstrained optimization problem with the Lagrangian: L(p,u,ψ t ) := Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x) + ψ t (x) ∂ t p t (x) +∇· p t (x)(f (x,t) + σ t u(x,t)) − σ 2 t ∆p t (x) dxdt(2.60) This optimization problem yields the pair of density and control drifts (u ⋆ ,p ⋆ t ) that satisfy the following optimality conditions (Caluya et al., 2021). Proposition 2.23 (Optimality Conditions for Dynamic Schr ̈odinger Bridge (Caluya et al., 2021)). The pair of optimal state PDF and optimal control (u ⋆ ,p ⋆ t ) that minimize (2.60) is the solution to the pair of PDEs defined as: ( ∂ t ψ t + σ 2 t 2 ∥∇ψ t ∥ 2 +⟨∇ψ t ,f⟩ =− σ 2 t 2 ∆ψ t ∂ t p ⋆ t +∇· (p ⋆ t (f + σ 2 t ∇ψ t )) = σ 2 t 2 ∆p ⋆ t s.t. ( p ⋆ 0 = π 0 p ⋆ T = π T (HJB-FP System) whereψ t (x) is the Lagrange multiplier. The optimal controlu ⋆ can also be written in terms of ψ t as: ∀(x,t)∈ R d × [0,T ], u ⋆ (x,t) = σ t ∇ψ t (x)(2.61) Proof. Now, we will break down the proof of this statement, which follows the proof of Proposition 1 in (Caluya et al., 2021). Expanding the Lagrangian from (2.60) we have: L(p,u,ψ) := Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x)dxdt + Z T 0 Z R d ψ t (x)∂ t p t (x)dxdt |z (⋆) + Z T 0 Z R d ψ t (x)∇· p t (x)(f (x,t) + σ t u(x,t)) dxdt | z (♦) − Z T 0 Z R d ψ t (x) σ 2 t 2 ∆p t (x)dxdt |z (▲) (2.62) Since we want to avoid derivatives with respect top ⋆ , we move the derivatives onto the Lagrange multiplier ψ t (x) using integration by parts. For the term (⋆), we have: Z R d Z T 0 ψ t (x) |z u ∂ t p t (x) | z dv dtdx = Z R d [ψ t (x)p t (x)] T t=0 | z constant − Z T 0 p t (x)∂ t ψ t (x)dt dx =− Z R d Z T 0 p t (x)∂ t ψ t (x)dtdx(2.63) where the boundary term reduces to constants R ψ 0 (x)π 0 (x)dtdxand R ψ T (x)π T (x)dtdx, which do not depend on the optimization variablep t and can be dropped from the Lagrangian. For the term (♦), we can apply integration by parts with the divergence identity R ψ t ∇·v=− R v·∇ψ t since the boundary condition vanishes at infinity 11 . Z T 0 Z R d ψ t (x)∇· p t (x)(f (x,t) + σ t u(x,t)) dxdt =− Z T 0 Z R d p t (x)∇ψ t (x)· (f (x,t) + σ t u(x,t))dxdt(2.64) 11 The marginal density p t must integrate to 1 and since R d is unbounded, it must vanish as ∥x∥→∞. 49 Finally, we apply integration by parts twice to simplify the term (▲): − Z T 0 Z R d ψ t (x) σ 2 t 2 ∆p t (x)dxdt =− Z T 0 Z R d ψ t (x) σ 2 t 2 d X i,j=1 ∂ 2 ∂x i ∂x j p t (x)dxdt =− d X i,j=1 Z T 0 σ 2 t 2 Z R d ψ t (x) | z u ∂ ∂x i ∂ ∂x j p t (x) |z dv dxdt = d X i,j=1 Z T 0 σ 2 t 2 Z R d ∂ ∂x i ψ t (x) |z u ∂ ∂x j p t (x) |z dv dxdt =− d X i,j=1 Z T 0 σ 2 t 2 Z R d p t (x) ∂ 2 ∂x i ∂x j ψ t (x)dxdt |z sum over i = j gives the Laplacian of ψ t =− Z T 0 Z R d σ 2 t 2 p t (x)∆ψ t (x)dxdt(2.65) Now, substituting all terms back into (2.62), we get: L(p t ,u,ψ) = Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x)dxdt− Z R d Z T 0 p t (x)∂ t ψ t (x)dtdx − Z T 0 Z R d p t (x)∇ψ t (x)· (f (x,t) + σ t u(x,t))dxdt− Z T 0 Z R d σ 2 t 2 p t (x)∆ψ t (x)dxdt = Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 − ∂ t ψ t (x)−∇ψ t (x)· (f (x,t) + σ t u(x,t))− σ 2 t 2 ∆ψ t (x) p t (x)dxdt(2.66) Isolating the terms that depend on u(x,t), we have the minimization problem: inf u 1 2 ∥u(x,t)∥ 2 −∇ψ t (x)· σ t u(x,t) |z (σ t u(x,t)) ⊤ (∇ψ t (x)) = 1 2 ∥u(x,t)∥ 2 − σ t u(x,t) ⊤ ∇ψ t (x) (2.67) By completing the square, we get: inf u 1 2 ∥u(x,t)∥ 2 − σ t u(x,t) ⊤ ∇ψ t (x)+ σ 2 t 2 ∥∇ψ t (x)∥ 2 | z square − σ 2 t 2 ∥∇ψ t (x)∥ 2 = inf u 1 2 ∥u(x,t)− σ t ∇ψ t (x)∥ 2 | z minimizing this yields u ⋆ − σ 2 t 2 ∥∇ψ t (x)∥ 2 (2.68) Since the second term doesn’t depend onu(x,t), minimizing the first term yields the expression (2.61) for the optimal u ⋆ (x,t): u ⋆ (x,t) = σ t ∇ψ t (x)(2.69) Substituting this expression into the Lagrangian in (2.62), we get: L(p,u ⋆ ,ψ) = Z T 0 Z R d 1 2 ∥σ t ∇ψ t (x) |z u ⋆ (x,t) ∥ 2 − ∂ t ψ t (x)−∇ψ t (x)· (f (x,t) + σ t σ t ∇ψ t (x) |z u ⋆ (x,t) )− σ 2 t 2 ∆ψ t (x) p t (x)dxdt = Z T 0 Z R d σ 2 t 2 ∥∇ψ t (x)∥ 2 − ∂ t ψ t (x)−⟨∇ψ t (x),f (x,t)⟩−σ 2 t ∥∇ψ t (x)∥ 2 − σ 2 t 2 ∆ψ t (x) p t (x)dxdt = Z T 0 Z R d − σ 2 t 2 ∥∇ψ t (x)∥ 2 − ∂ t ψ t (x)−⟨∇ψ t (x),f (x,t)⟩− σ 2 t 2 ∆ψ t (x) p t (x)dxdt(2.70) 50 Sinceu ⋆ determines the optimalp ⋆ t via the Fokker-Planck constraint, the Lagrangian cannot vary with respect top ⋆ t , so the expression within the curly brackets must vanish for optimality to hold over arbitrary p t : − σ 2 t 2 ∥∇ψ t (x)∥ 2 − ∂ t ψ t (x)−⟨∇ψ t (x),f (x,t)⟩− σ 2 t 2 ∆ψ t (x) = 0(2.71) or equivalently: ∂ t ψ t (x) + σ 2 t 2 ∥∇ψ t (x)∥ 2 +⟨∇ψ t (x),f (x,t)⟩ = σ 2 t 2 ∆ψ t (x)(2.72) which is exactly the Hamilton-Jacobi-Bellman (HJB) equation in (HJB-FP System). Substituting the optimal expression foru ⋆ (x,t) into the FP equation in (HJB-FP System), we obtain the FP equation for the optimal density function p ⋆ t : ∂ t p ⋆ t (x) =−∇· p ⋆ t (x)(f (x,t) + σ t σ t ∇ψ t (x) | z u ⋆ (x,t) ) + σ 2 t 2 ∆p ⋆ t (x)(2.73) which gives the Fokker-Planck (FP) equation (HJB-FP System). Finally, the boundary conditions p ⋆ 0 =π 0 andp ⋆ T =π T follow directly from (Density Dynamic SB Problem), which yields the system of non-linear PDEs: ( ∂ t ψ t + σ 2 t 2 ∥∇ψ t ∥ 2 +⟨∇ψ t ,f⟩ =− σ 2 t 2 ∆ψ t ∂ t p ⋆ t +∇· (p ⋆ t (f + σ 2 t ∇ψ t )) = σ 2 t 2 ∆p ⋆ t s.t. ( p ⋆ 0 = π 0 p ⋆ T = π T (HJB-FP System) which define the optimal pair (ψ t ,p ⋆ t ) and equivalently (u ⋆ ,p ⋆ t ) that solve the dynamic SB problem. □ Although the coupled nonlinear PDE system in (HJB-FP System) fully characterizes the optimal control–density pair (u ⋆ ,p ⋆ t ), solving this system remains challenging in general, particularly under nonlinear prior dynamics and arbitrary marginal constraints. A key step toward tractability is the Hopf–Cole transform, which converts the nonlinear system in (HJB-FP System) into an equivalent pair of linear PDEs. Before moving on to deriving these linear PDE optimality conditions, let’s first briefly review several commonly used reference processesQthat appear across SB formulations and modern generative modeling frameworks. Reference Processes for Schr ̈odinger Bridges We now define a few common reference processesQ, in addition to pure Brownian motionσB, to reveal the connection between various generative modeling paradigms and the dynamic Schr ̈odinger bridge problem. Ornstein–Uhlenbeck (OU) processes. The class of Ornstein–Uhlenbeck (OU) pro- cesses is a common form of reference process in SB literature (De Bortoli et al., 2021; Shi et al., 2023; Vargas et al., 2021) and takes the form: dX t =−βX t dt + σ t dB t (Ornstein–Uhlenbeck SDE) Unlike pure Brownian motion, the OU process introduces a deterministic drift−βX t that pulls the system toward the origin, creating mean-reverting stochastic processes. While Brownian motion diverges in variance, OU processes approach a stationary Gaussian distribution. 51 Variance Exploding SDEs. The classic score-based generative modeling framework (Huang et al., 2021; Song et al., 2019; 2020b) adopts the class of variance exploding SDEs (VESDEs) as the reference process, which takes the form: dX t = q dσ 2 t /dtdB t , X 0 ∼ π 0 (Variance Exploding SDE) whereσ 2 t is the variance that increases with time. By integrating the SDE over time [0,t], the variance is defined asβ t := R t 0 σ 2 s ds, which increases over time. Therefore, the marginal density is given by N (0,β t I d ). Variance Preserving SDEs. Another class of reference processes are the variance- preserving SDEs (VPSDEs) that underlie the widely-adopted denoising diffusion prob- abalistic model (DDPM) framework (Ho et al., 2020; Sohl-Dickstein et al., 2015; Song et al., 2020b), which take the form: dX t =− 1 2 β t X t dt + p β t dB t , X 0 ∼ π 0 (Variance Preserving SDE) Unlike VESDEs, the drift and diffusion terms in VPSDEs are carefully balanced so that the total variance of the process remains approximately constant, and the marginal distribution follows a constant-variance Gaussian N (0,I d ). Other reference processes have also been explored, including alternative definitions of the coefficientσ t in VESDEs (Karras et al., 2022; Song et al., 2023), sub-VPSDEs (Song et al., 2020b), and fractional Brownian motion (Nobis et al., 2025), which we will explore in depth in Section 5.6. Having defined the dynamic SB problem given an arbitrary reference processQ, the system of non-linear PDEs that define the optimal control-density pair (u ⋆ ,p ⋆ t ), and some examples of reference processesQcommonly seen in literature, we will move on to the derivation of the Hopf-Cole transform. This transform converts the nonlinear PDE conditions to a system of linear PDE conditions, where the solution is a pair of time-evolving Schr ̈odinger potentials (φ t , ˆφ t ) that define the marginal density of the SB, which are analogous to the static Schr ̈odinger potentials (φ, ˆφ) that define the optimal static SB problem π ⋆ 0,T from Section 1.4. 2.8 Hopf-Cole Transform The Hopf-Cole transform allows us to transform the non-linear PDEs coupled over the full trajectory into a system of linear PDEs that are only coupled via their boundary constraints. This is achieved with a change-of-variables from the optimal control–density pair (u ⋆ ,p ⋆ t ) to a pair of potential functions (φ t , ˆφ t ) that define a coupled system of linear PDEs. To derive this, we first recall the (HJB-FP System) as: ∂ t ψ t + σ 2 t 2 ∥∇ψ t ∥ 2 +⟨∇ψ t ,f⟩ =− σ 2 t 2 ∆ψ t (HJB Equation) ∂ t p ⋆ t +∇· (p ⋆ t (f + σ 2 t ∇ψ t )) = σ 2 t 2 ∆p ⋆ t (FP Equation) To understand the intuition behind the Hopf-Cole transform, we can make the simple observation that the quadratic term σ 2 t 2 ∥∇ψ∥ 2 is the only non-linear term in the HJB-FP system. Therefore, the goal of the transform is to answer: how do we define a change-of-variables that makes this linear? 52 That is exactly the goal of the Hopf-Cole Transform, which defines a change-of-variables to the non-linear HJB-FP system such that they can be solved via a system of linear PDEs, whose optimality conditions align exactly to the solution of the non-linear system. Theorem 2.24 (Hopf-Cole Transform). Given an reference process defined by the deterministic driftf(x,t), diffusion coefficientσ t , and boundary marginal distributionsπ 0 ,π T ∈P(R d ), we can apply the following change of variables (ψ,p ⋆ t )7→ (φ t , ˆφ t ) defined as: ψ t (x) = logφ t (x), p ⋆ t (x) = φ(x) ˆφ(x)(Hopf-Cole Transform) which transforms (HJB Equation) and (FP Equation) into a system of linear PDEs for (φ t , ˆφ t ) given by: ( ∂ t φ t +⟨∇φ t ,f⟩ =− σ 2 t 2 ∆φ t ∂ t ˆφ t +∇· ( ˆφ t f ) = σ 2 t 2 ∆ ˆφ t s.t. ( p ⋆ 0 = φ 0 ˆφ 0 p ⋆ T = φ T ˆφ T (Hopf-Cole PDEs) which define the dynamics and terminal conditions of (φ t , ˆφ t ). Furthermore, the optimal control can be written as: u ⋆ (x,t) = σ t ∇ logφ t (x)(Optimal Control with SB Potential) Derivation. To derive this from first principles, we can start with a well-known property of the Laplacian operation, which is that the Laplacian of a logarithm produces squared gradients. We can leverage this property to write the Laplacian ∆ψ t as a squared gradient to cancel out the quadratic term. Specifically, let us define an ansatz for the change of variables ψ 7→ φ as: ψ t (x) = C logφ t (x) ⇐⇒ φ t (x) = exp ψ t (x) C (2.74) where C is some unknown scalar constant. We now express each term containing ψ t in terms of φ. (i) The quadratic term becomes: σ 2 t 2 ∥∇ψ t ∥ 2 = σ 2 t 2 ∥∇(C logφ t )∥ 2 = σ 2 t C 2 2 ∇φ t φ t 2 = σ 2 t C 2 2 ∥∇φ t ∥ 2 φ 2 t (2.75) (i) The Laplacian term becomes: − σ 2 t 2 ∆ψ t =− σ 2 t 2 (∇·∇(C logφ t )) =− σ 2 t 2 C∇· ∇φ t φ t =− σ 2 t 2 C∇· φ −1 t ∇φ t (2.76) Applying the product rule of divergences ∇· (uv) = (∇u)·v + u(∇·v), we get: − σ 2 t 2 ∆ψ =− σ 2 t 2 C ∇· φ −1 ∇φ =− σ 2 t 2 C ∇(φ −1 )·∇φ + φ −1 (∇·∇φ) (2.77) =− σ 2 t 2 C ∆φ φ − ∥∇φ∥ 2 φ 2 = σ 2 t 2 C ∥∇φ∥ 2 φ 2 − ∆φ φ (2.78) To determine the value ofCthat allows the quadratic term to cancel, we set them equal to each other: σ 2 t C 2 2 ∥∇φ∥ 2 φ 2 = σ 2 t 2 C ∥∇φ∥ 2 φ 2 =⇒ C = 1(2.79) 53 Now, we have defined the mappingψ t 7→ φ t asψ t (x) =logφ t (x) which makes the HJB PDE linear. Substituting ψ t (x) = logφ t (x) into the (FP Equation), we get: ∂ t logφ t +⟨∇ logφ t ,f⟩ + σ 2 t 2 ∥∇ logφ t ∥ 2 =− σ 2 t 2 ∆ logφ t 1 φ t ∂ t φ t + 1 φ t ⟨∇φ t ,f⟩ + σ 2 t 2 ∥∇φ t ∥ 2 φ 2 t =− σ 2 t 2 ∆φ t φ t − ∥∇φ t ∥ 2 φ 2 t 1 φ t ∂ t φ t + 1 φ t ⟨∇φ t ,f⟩ + σ 2 t 2 ∥∇φ t ∥ 2 φ 2 t =− σ 2 t 2 ∆φ t φ t + σ 2 t 2 ∥∇φ t ∥ 2 φ 2 t (2.80) Subtracting σ 2 t 2 ∥∇φ∥ 2 φ 2 from both sides and multiplying by φ t , we get: ∂ t φ t +⟨∇φ t ,f⟩ =− σ 2 t 2 ∆φ t (2.81) which is exactly the first equation in (Hopf-Cole PDEs). Now, we need to define a change-of- variables forp t that decouplesp ⋆ t and∇ψ t in the (FP Equation). Substituting our first mapping ψ t (x) = logφ t (x) into the Fokker-Planck equation, we get: ∂ t p ⋆ t +∇· (p ⋆ t (f + σ 2 t ∇ψ t )) = σ 2 t 2 ∆p ⋆ t ∂ t p ⋆ t +∇· (p ⋆ t (f + σ 2 t ∇ x logφ t )) = σ 2 t 2 ∆p ⋆ t (2.82) Observing this equation, we see that the term∇ logφ t appears as an additional drift, which reweights the trajectories according to the potentialφ t , but also couples the densityp ⋆ t with the potential φ t . To decouple this interaction, we can divide the density byφ t to get the change of variables ˆφ t (x) = p ⋆ t (x) φ t (x) such that the optimal density factorizes as: ˆφ t (x) = p ⋆ t (x) φ t (x) =⇒ p ⋆ t (x) = ˆφ t (x)φ t (x) = ˆφ t (x)e ψ t (x) (2.83) which separates the dynamics into forward and backward components. Substitutingp ⋆ t =φ t ˆφ t into each term in the (FP Equation), we get: (i) The time derivative becomes: ∂ t (φ t ˆφ t ) = (∂ t φ t ) ˆφ t + φ t (∂ t ˆφ t )(2.84) and substituting the expression ∂φ ∂t =−⟨∇φ t ,f⟩− σ 2 t 2 ∆φ t from the first PDE, we get: ∂ t p ⋆ t = −⟨∇φ t ,f⟩− σ 2 t 2 ∆φ t ˆφ t + φ t (∂ t ˆφ t ) =− ˆφ t ⟨∇φ t ,f⟩− σ 2 t 2 ˆφ t ∆φ t + φ t (∂ t ˆφ t )(2.85) (i) The drift term becomes: ∇·(p ⋆ t (f + σ 2 t ∇ x logφ t )) =∇· (φ t ˆφ t (f + σ 2 t ∇ x logφ t )) =∇· φ t ˆφ t f + σ 2 t ∇ x φ t φ t =∇· (φ t ˆφ t f ) +∇· σ 2 t (φ t ˆφ t ) ∇ x φ t φ t =∇· (φ t ˆφ t f ) + σ 2 t ∇· ( ˆφ t ∇ x φ t ) =⟨∇(φ t ˆφ t ),f⟩ + (φ t ˆφ t )∇·f + σ 2 t (∇ ˆφ t ·∇φ t + ˆφ t ∆φ t ) = ˆφ t ⟨∇φ t ,f⟩ + φ t ⟨∇ ˆφ t ,f⟩ + (φ t ˆφ t )∇·f + σ 2 t ∇ ˆφ t ·∇φ t + σ 2 t ˆφ t ∆φ t (2.86) 54 (i) The Laplacian term becomes: σ 2 t 2 ∆p ⋆ t = σ 2 t 2 ∆(φ t ˆφ t ) = σ 2 t 2 ∇· ( ˆφ t ∇φ t + φ t ∇ ˆφ t ) = σ 2 t 2 (∇· ( ˆφ t ∇φ t ) +∇· (φ t ∇ ˆφ t )) = σ 2 t 2 (∇ ˆφ t ·∇φ t + ˆφ∆φ +∇φ t ·∇ ˆφ + φ t ∆ ˆφ t ) = σ 2 t 2 ( ˆφ t ∆φ t + 2∇ ˆφ t ·∇φ t + φ t ∆ ˆφ t ) = σ 2 t 2 ˆφ t ∆φ t + σ 2 t ∇ ˆφ t ·∇φ t + σ 2 t 2 φ t ∆ ˆφ t (2.87) Putting it together and canceling terms, we get: ∂ t p ⋆ t +∇· (p ⋆ t ∇ logφ t ) = σ 2 t 2 ∆p ⋆ t − ˆφ t ⟨∇φ t ,f⟩− σ 2 t 2 ˆφ t ∆φ t + φ t (∂ t ˆφ t ) |z (2.85) + ˆφ t ⟨∇φ t ,f⟩ + φ t ⟨∇ ˆφ t ,f⟩ + (φ t ˆφ t )∇·f + σ 2 t ∇ ˆφ t ·∇φ t + σ 2 t ˆφ t ∆φ t |z (2.86) = σ 2 t 2 ˆφ t ∆φ t + σ 2 t ∇ ˆφ t ·∇φ t + σ 2 t 2 φ t ∆ ˆφ t | z (2.87) φ t (∂ t ˆφ t ) + φ t ⟨∇ ˆφ,f⟩ + (φ t ˆφ t )∇·f = σ 2 t 2 φ t ∆ ˆφ t ∂ t ˆφ t +⟨∇ ˆφ,f⟩ + ˆφ t ∇·f | z =∇·(f ˆφ t ) = σ 2 t 2 ∆ ˆφ t ∂ t ˆφ t +∇· ( ˆφ t f ) = σ 2 t 2 ∆ ˆφ t (2.88) which recovers the second equation in (Hopf-Cole PDEs). Now, we have derived the Hopf-Cole transform (ψ,p ⋆ t )7→ (φ t , ˆφ t ) that solve a system of linear PDE equations: ( ∂ t φ t +⟨∇φ t ,f⟩ =− σ 2 t 2 ∆φ t ∂ t ˆφ t +∇· ( ˆφ t f ) = σ 2 t 2 ∆ ˆφ t (Hopf-Cole PDEs) which uniquely define the optimal control-density pair (u ⋆ ,p ⋆ t ) of the Schr ̈odinger bridge: u ⋆ (x,t) = σ t ∇ logφ t (x), p ⋆ t (x) = φ t (x) ˆφ t (x) (Optimal Control-Density Pair) yielding the boundary constraintsp ⋆ 0 (x) =π 0 (x) =φ 0 (x)ˆφ 0 (x) andp ⋆ T (x) =π T (x) =φ T (x)ˆφ T (x). □ Given these optimality constraints, we can derive a system of equations that explicitly define the forward and backward potential functions. Corollary 2.25 (Forward–Backward Schr ̈odinger Potentials). Let (φ t , ˆφ t ) denote the Schr ̈odinger potentials obtained through the Hopf-Cole transform, which define the solution to the (Dynamic OT Problem) as: p ⋆ t (x) = φ t (x) ˆφ t (x), u ⋆ (x,t) = σ t ∇ logφ t (x)(2.89) Then, (φ t , ˆφ t ) can be represented as the solution to a system of equations with the transition density under the reference path measure Q given by: ( φ t (x) = R R d Q T|t (y|x)φ T (y)dy ˆφ t (x) = R R d Q t|0 (x|y) ˆφ 0 (y)dy s.t. ( π 0 (x) = φ 0 (x) ˆφ 0 (x) π T (x) = φ T (x) ˆφ T (x) (Schr ̈odinger Potentials) 55 subject to the boundary factorization constraints. Proof. Observe that the system of linear PDEs derived in Theorem 2.24 aligns with the form of the (Kolmogorov Backward Equation) defined in Corollary 2.16, which we recall is defined for some function r(x,t) as: ∂ t r(x,t) +⟨f (x,t),∇r(x,t)⟩ + σ 2 t 2 ∆r(x,t) = 0, r(x,T ) = Φ(x)(Kolmogorov Backward Equation) which admits the (Feynman-Kac Representation): r(x,t) := E X t:T ∼Q [Φ(X T )|X t = x](Feynman-Kac Representation) The forward potentialφ t satisfies the first linear PDE in (Hopf-Cole PDEs), which is equivalent to the backward Kolmogorov equation withr(x,t) :=φ t (x) and Φ(x) =φ T (x) associated with the reference process Q. Therefore, φ t can be written with the Feynman-Kac representation as: φ t (x) = E X t:T ∼Q [φ T (X T )|X t = x](2.90) Equivalently, since the conditional density ofX T givenX t =xis defined by the transition density under the reference processQ T|t (·|x) =Q(X T =·|X t =x), we can write the Feynman-Kac representation the linear integral operator: φ t (x) = Z R d Q T|t (x T |x)φ T (x T )dx T (Forward Potential) Similarly, the backward potentialˆφ t satisfies the second linear PDE in (Hopf-Cole PDEs), which is equivalent to the backward Kolmogorov equation withr(x,t) :=ˆφ t (x) and terminal constraint Φ(x) = ˆφ 0 (x). Therefore, ˆφ t can be written with the Feynman-Kac representation as: ˆφ t (x) = E X 0:t ∼Q [ ˆφ 0 (X 0 )|X t = x] = Z R d Q t|0 (x|x 0 ) ˆφ 0 (x 0 )dx 0 (Backward Potential) which are exactly the equations defined in (Schr ̈odinger Potentials).□ This result shows that the forward potentialφ t propogates backward from the terminal constraint π T =φ T ˆφ T via a linear integral operator over the expected distribution ofφ T (X T ) and the backward potential propogates forward from the reversed terminal constraintπ 0 =φ 0 ˆφ 0 via a linear integral operator in the reverse time over the expected distribution of ˆφ 0 (X 0 ). Recall from our discussion on the static Schr ̈odinger bridge problem in Section 1.4 where we define the pair of SB potentials (φ, ˆφ) as the solution to the (Schr ̈odinger System). Crucially, we can interpret the Schr ̈odinger potentials (φ t , ˆφ t ) obtained through the Hopf–Cole transform as the dynamic or continuous-time analogues of the Schr ̈odinger potentials. In the static formulation, the optimal coupling between the marginals (π 0 ,π T ) under the reference kernelK(x,y) =e −c(x,y) admits the factorized form: π ⋆ 0,T (x,y) = e φ(x)+ ˆφ(y)−c(x,y) = e φ(x) K(x,y)e ˆφ(y) (Solution to Static SB) The (Dynamic SB) problem generalizes this structure from couplings of endpoints to path measures of stochastic processes. In this setting, the time-dependent potentials (φ t , ˆφ t ) propagate according to the forward–backward linear PDE system derived above, and their product recovers the optimal marginal density at each time p ⋆ t = φ t ˆφ t . As shown in Theorem 1.10, the static Schr ̈odinger system admits a unique solution up to an additive constant. Since the solution to the dynamic SB problemp ⋆ t is also unique by the strict convexity of the KL divergence, the productφ t ˆφ t is unique, and the individual potentialsφ t andˆφ t are unique up to a multiplicative constant which leaves their product invariant. 56 2.9 Schr ̈odinger Bridges as Entropy-Regularized Dynamic Optimal Transport One of the most important perspectives on the Schr ̈odinger bridge problem is its close relationship to optimal transport. In Section 1, we derived the static SB problem directly from the entropic OT problem, with a simple reparameterization of the reference coupling. In this section, we make a similar connection to the Benamou and Brenier (dynamic) formulation of the OT problem defined in Section 2.1. We will show that by reparameterizing the control drift in the controlled Fokker–Planck equation, the stochastic dynamics can be expressed as a deterministic continuity equation describing the transport of probability mass by a velocity field. Under this transformation, the dynamic SB objective decomposes into a kinetic transport energy that aligns with the dynamic OT objective, with additional entropy-regularization terms induced by diffusion. Proposition 2.26 (Dynamic Optimal Transport Form of Schr ̈odinger Bridge). Consider the dynamic Schr ̈odinger bridge problem written in terms of the marginal density in (Controlled Fokker-Planck Equation) as: inf (u,p t ) " Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x)dxdt # (Density Dynamic SB Objective) s.t. ( ∂ t p t (x) =−∇· p t (x)(f (x,t) + σ t u(x,t)) + σ 2 t 2 ∆p t (x) p 0 = π 0 , p T = π T which enforces the marginal constraintsp 0 =π 0 andp T =π T and Fokker-Planck equation constraint. Then, by reparameterizingv(x,t) :=u(x,t) + σ t 2 ∇ logp t (x), the dynamic SB problem takes an equivalent form of a dynamic optimal transport problem given by: inf v E p t Z T 0 1 2 ∥v(x,t)∥ 2 + σ 2 t 8 ∥∇ logp t (x)∥ 2 − 1 2 ⟨∇ logp t (x),f (x,t)⟩ dt (Entropy-Regularized Dynamic OT) ( ∂ t p t (x) =−∇· (p t (f (x,t) + σ t v(x,t))) p 0 = π 0 , p T = π T where ∂ t p t =−∇· (p t (f t + σ t v)) is the continuity equation constraint. Proof. In this proof, our goal is to absorb the stochastic term in the Fokker-Planck constraint into the objective functional, such that the constrained objective is in the form of the (Continuity Equation), which can be interpreted as a special case of the Fokker-Planck equation (Remark 2.13). Step 1: Rewrite Fokker-Planck Equation as Continuity Equation. We start by rewriting the Fokker-Planck equation by absorbing the Laplacian diffusion term into the divergence term to 57 get the continuity equation with a newly defined drift field v(x,t): ∂ t p t (x) =−∇· p t (x)(f (x,t) + σ t u(x,t)) + σ 2 t 2 ∆p t (x) =−∇· p t (x)(f (x,t) + σ t u(x,t)) + σ 2 t 2 ∇·(∇p t (x)) =−∇· p t (x)(f (x,t) + σ t u(x,t))− σ 2 t 2 ∇p t (x) =−∇· p t (x) f (x,t) + σ t u(x,t)− σ 2 t 2 ∇ logp t (x) =−∇· p t (x) f (x,t) + σ t u(x,t)− σ t 2 ∇ logp t (x) |z =:v(x,t) =−∇· (p t (x)(f (x,t) + σ t v(x,t))(Continuity Equation) where we define a new velocityv(x,t) :=u(x,t)− σ t 2 ∇ logp t (x) that satisfies a continuity equation. We rearrange to write u(x,t) in terms of the new velocity v(x,t) as: v(x,t) = u(x,t)− σ t 2 ∇ logp t (x) =⇒ u(x,t) = v(x,t) + σ t 2 ∇ logp t (x)(2.91) Substituting this expression (2.91) foru(x,t) into the objective functional for the (Density Dynamic SB Objective), we get: Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x)dxdt = Z T 0 Z R d 1 2 v(x,t) + σ t 2 ∇ logp t (x) 2 p t (x)dxdt = Z T 0 Z R d 1 2 ∥v(x,t)∥ 2 + 2 D v(x,t), σ t 2 ∇ logp t (x) E + σ 2 t 4 ∥∇ logp t (x)∥ 2 p t (x)dxdt = Z T 0 Z R d 1 2 ∥v(x,t)∥ 2 |z kinetic energy + σ t 2 ⟨v(x,t),∇ logp t (x)⟩ |z cross term + σ 2 t 8 ∥∇ logp t (x)∥ 2 |z Fisher information p t (x)dxdt(2.92) which is the objective that corresponds to the (Continuity Equation). Since the kinetic energy 1 2 ∥v(x,t)∥ 2 is convex inv, we want to write the objective such that the drift being optimized (i.e., v) appears only in the kinetic energy. To do this, we will rewrite the cross term dependent onvin the next step. Step 2: Expand the Cross-Term. Observing (2.92), the cross term is dependent on both of the variables being optimizedvandp t , which yields challenges in practice. Isolating the cross term, we aim to expand it to remove the cross dependency: Z T 0 Z R d σ t 2 ⟨v(x,t),∇ logp t (x)⟩p t (x)dxdt(Cross Term) Intuitively, we can consider the expression⟨v,∇ logp t ⟩p t =v·∇p t as measuring the alignment between the velocity fieldvand the gradient of the density. In other words, it answers: is velocity field pushing probability mass toward regions of high density or away from them? This has a direct connection to entropy, which measures the concentration of probability density. LetH(p t ) = R p t logp t dxdenote the entropy. Since the objective integrates the cross term over timet∈[0,T], we consider the change 58 in entropy between the terminal marginals: H(p T )− H(p 0 ) = Z T 0 ∂ t H(p t )dt = Z T 0 ∂ t Z R d p t logp t dx dt = Z T 0 Z R d logp t ∂ t p t + p t ∂ t logp t dxdt = Z T 0 Z R d (1 + logp t )∂ t p t dxdt(2.93) Substituting the continuity equation constraint ∂ t p t =−∇· (p t (f + σ t v)), we have: H(p T )− H(p 0 ) = Z T 0 Z R d (1 + logp t ) −∇· (p t (f + σ t v) dxdt(2.94) Then, applying the integration of parts identity for divergence 12 , we can rewrite (2.94) as: H(p T )− H(p 0 ) = Z T 0 Z R d ∇ logp t ,p t (f + σ t v) dxdt = Z T 0 Z R d ∇ logp t , (f + σ t v) p t dxdt = Z T 0 Z R d ∇ logp t ,f p t dxdt + Z T 0 Z R d σ t ∇ logp t ,v p t dxdt =⇒ Z T 0 Z R d σ t 2 ∇ logp t ,v p t dxdt = 1 2 (H(p T )− H(p 0 )) |z constant − Z T 0 Z R d 1 2 ∇ logp t ,f p t dxdt (2.95) In the (Density Dynamic SB Objective), the marginal distributionsp 0 =π 0 andp T =π T are fixed, so the entropy difference termH(p T )− H(p 0 ) is a constant and can be dropped in the objective. Therefore, substituting (2.94) into the objective functional, we have: inf (p t ,v,g) Z T 0 Z R d 1 2 ∥v(x,t)∥ 2 + σ 2 t 8 ∥∇ logp t (x)∥ 2 − 1 2 ⟨∇ logp t (x),f (x,t)⟩ p t (x)dxdt(2.96) Equivalently, we can write (2.96) as an expectation over the marginal densityp t corresponding to the velocity field v t given by: inf v E p t Z T 0 1 2 ∥v(x,t)∥ 2 + σ 2 t 8 ∥∇ logp t (x)∥ 2 − 1 2 ⟨∇ logp t (x),f (x,t)⟩ dt(2.97) ( ∂ t p t (x) =−∇· (p t (f (x,t) + σ t v(x,t))) p 0 = π 0 , p T = π T (2.98) which concludes our derivation of the dynamical OT form of the dynamic SB problem.□ This derivation yields an objective with three distinct terms that capture both the transport and diffusion terms in the original Fokker-Planck constraint. Intuitively, each term can be interpreted as follows: (i)The kinetic energy term 1 2 ∥v (x,t)∥ 2 measures the deterministic cost of transporting proba- bility mass along the velocity fieldv(x,t). This term is exactly the kinetic energy appearing in the (Dynamic OT Problem), where the optimal velocity minimizes the kinetic energy required to move mass on the deterministic flow between the marginal distributions. 12 which states for a scalar function φ and vector ξ, the following holds: R φ(−∇·ξ)dx = R ⟨∇φ,ξ⟩dx 59 (i)The Fisher information term σ 2 t 4 ∥∇ logp t (x)∥ 2 measures the sharpness of the marginal distribution. Smooth distributions yield low Fisher information, while sharp and concentrated distributions yield high Fisher information. This term appears from the diffusion term in the Fokker-Planck equation as the diffusion in the SDE acts to smooth out the distribution. (i)The cross term or drift interaction 1 2 ⟨∇ logp t (x),f(x,t)⟩measures the interaction between the reference driftf(x,t) and the density evolution∇ logp t . This term reflects how the prior dynamics influence the evolution of the distribution by either aligning with or opposing the natural directions of increasing probability mass. This result shows that the (Dynamic SB Problem) problem can be explicitly rewritten as an entropy- regularized version of the (Dynamic OT Problem), where the optimization over stochastic path measuresP u reduces to an optimization over deterministic density flowsp t and velocity fieldsv(x,t). In the limit of vanishing diffusion (σ t →0), the entropy regularization from the Fisher information and cross-term vanish, and the formulation recovers the classic (Dynamic OT Problem). 2.10 Closing Remarks for Section 2 In this section, we took the crucial step of lifting the static formulation of the Schr ̈odinger bridge (SB) problem to the space of continuous-time path measures. Starting from the dynamic optimal transport (OT) problem, we show that the static OT problem with a quadratic cost function can be written in an equivalent dynamic form, which aims to find the optimal probability flow that smoothly transports mass between the prescribed marginals via a velocity field over a continuous time interval, yielding the straight probability flow between distributions. Given that most real-world systems do not evolve in straight lines between distributions, we extend the entropic OT problem to the space of stochastic processes, where path measures are determined by both a deterministic drift and random fluctuations in the form of Brownian motion, and entropy regularization is performed with a reference stochastic process. To understand the dynamic SB problem, we first introduce the governing theory of path measures and Itˆo processes, from Itˆo’s formula, which describes the evolution of functions evaluated on stochastic processes, to the Fokker-Planck and Feynman-Kac equations, which characterize how probability densities and functions evolve over stochastic processes through partial differential equations (PDEs). Just like the static case, we introduce the concept of relative entropy in the space of path measures, which can be expressed explicitly in terms of the difference between the drifts of two stochastic differential equations using Girsanov’s theorem. Building on this framework, we analyzed the dynamic SB problem by expressing the KL minimization over path measures as a minimization of the kinetic energy of a control drift, corresponding to the minimal perturbation required to steer the reference SDE so that its marginals match the prescribed constraints. We leverage Lagrange multipliers to derive the optimality conditions for the dynamic SB problem, which result in a pair of non-linear PDEs describing the forward and backward dynamics of the optimal solution (ψ t ,p ⋆ t ). Finally, we introduce the Hopf-Cole transform as a method of linearizing the non-linear PDEs with a simple change-of-variables, revealing that the optimal SB dynamics can be factorized into the product of a pair of forward and backward potentials p ⋆ t = φ t ˆφ t which solve a pair of linear PDEs. While we have analyzed the dynamic SB problem through the lenses of minimizing the KL divergence between stochastic path measures, it admits an alternative interpretation as an optimal control problem, where the goal is to determine the control drift that minimizes the expected future cost of steering a stochastic system toward a desired terminal distribution. This is precisely the idea of stochastic optimal control (SOC), which leverages Bellman’s Principle of Optimality to define an optimal control as the minimizer to the expected cost-to-go from an intermediate state to all 60 possible terminal states under an SDE. In the next section, we explore this perspective in detail, first introducing the theory of SOC and its properties before explicitly reformulating the dynamic SB problem as an SOC problem. 61 3. Schr ̈odinger Bridge Problem as Optimal Control In this section, we reformulate the Schr ̈odinger bridge (SB) problem through the lens of stochastic optimal control (SOC) theory, providing a dynamic and decision-theoretic perspective on entropy- regularized transport. Rather than directly optimizing over path measures, this view interprets the SB problem as learning an optimal control that steers a reference stochastic process between prescribed marginals while minimizing a control cost. We begin in Section 3.1 by introducing the general SOC framework, including Bellman’s Principle of Optimality and the value function that characterizes optimal policies under terminal constraints. In Section 3.2, we establish the connection between SB and SOC by showing that the optimal bridge corresponds to an optimal control drift applied to the reference dynamics. Finally, in Section 3.3, we develop practical training objectives that enable solving the resulting control problem in high-dimensional settings. Since we work with multiple path measures with different controls, we denote the stochastic process generated with a specific controluwith the superscripted notation X u 0:T = (X u t ) t∈[0,T ] to clearly denote the path measure P u that the process was generated under. 3.1 Stochastic Optimal Control While the Schr ̈odinger bridge problem aims to optimize the intermediate bridge between fixed endpoints, it is natural to consider an alternative variational perspective where given a particle sampled from the initial distributionX 0 ∼ π 0 , we want to optimize its path such that it reaches a state in the target distribution while minimially deviating from the reference SDE. This is the key idea behind stochastic optimal control (SOC) theory (N ̈usken et al., 2021), which seeks an optimal control driftu ⋆ (x,t) that corrects the particles trajectory such that it takes the path of minimal cost toward the target distribution π T . Definition 3.1 (Stochastic Optimal Control (SOC) Objective). Given an running cost c(x,t) :R d ×[0,T]→ Rand a terminal cost Φ(x) :R d → R, we consider the following stochastic optimal control (SOC) objective: inf u E X u 0:T ∼P u " Z T 0 1 2 ∥u(X u t ,t)∥ 2 + c(X u t ,t) dt + Φ(X u T ) # (SOC Objective) s.t. dX u t = (f (X u t ,t) + σ t u(X u t ,t))dt + σ t dB t , X u 0 ∼ π 0 whereu(x,t) is the control drift which produces the path measureP u ,f(x,t) is the drift of the reference processQ,σ t is the diffusion coefficient, andB t isd-dimensional Brownain motion. Under this objective, we can define the cost functionalJ(x,t;u) being optimized as the cost-to-go from any fixed point (x,t)∈ R d ×[0,T] at timetunder the controluas the expected running cost and terminal cost of integrating the controlled SDE from X u t = x over s∈ [t,T ]. J (x,t;u) := E X u t:T ∼P u Z T t 1 2 ∥u(X u s ,s)∥ 2 + c(X u s ,s) ds + Φ(X u T ) X u t = x (Cost Functional) Given the cost-to-go for an arbitrary controlu, we define the value functionV t (x) :R d → Ras the optimal cost-to-go obtained with the optimal control u ⋆ . V t (x) := J ⋆ (x,t;u ⋆ ) := inf u J (x,t;u)(Value Function) which solves the Hamilton-Jacobi-Bellman (HJB) equation, similarly to the Lagrange multiplier ψ t (x) from Section 2.7. Before defining the HJB for the value function, we first define Bellman’s 62 Principle of Optimality, also referred to as the dynamic programming principle, which provides an intuitive and rigorous foundation for stochastic optimal control theory. Definition 3.2 (Bellman’s Principle of Optimality). LetV t (x) denote the value function or optimal cost-to-go of a stochastic control problem starting from statexat timet. Then, the optimal control satisfies Bellman’s principle of optimality, which states that for all intermediate time stepst≤ τ ≤ Tthe optimal cost-to-go is equal to the cost incurred over the interval [t,τ] and the future cost-to-goV τ (X τ ) starting fromX τ over time [τ,T], which yields the following expanded expression or the value function: V t (x) = inf u E X u t:T ∼P u Z τ t + Z T τ 1 2 ∥u(X u s ,s)∥ 2 + c(X u s ,s) ds + Φ(X u T ) X u t = x = inf u E X u t:T ∼P u Z τ t 1 2 ∥u(X u s ,s)∥ 2 + c(X u s ,s) ds + V τ (X u τ ) X u t = x (Bellman’s Principle of Optimality) Intuitively, this means that in an optimally controlled process, regardless of the initial statexand changes over [t,τ], the remaining controlled process fromX τ follows the optimal control law for the state resulting from the first decision. Taking the infinitesimal limit whenτ → tandτ − t→0, this property yields the Hamilton–Jacobi–Bellman (HJB) equations (Bardi et al., 1997), which characterizes the evolution of the value function. Lemma 3.3 (Hamilton-Jacobi-Bellman (HJB) Equations). Given the infinitesimal generator A t of the uncontrolled SDEdX t =f(X t ,t)dt+σ t dB t that acts on the value functionV t (x) as: (A t V t )(x) :=⟨f (x,t),∇V t (x)⟩ + σ 2 t 2 ∆V t (x)(3.1) Then, V t (x) solves the Hamilton-Jacobi-Bellman equation defined as: ∂ t V t (x) =−(A t V t )(x) + σ 2 t 2 ∥∇V t (x)∥ 2 − c(x,t), V T (x) = Φ(x)(Value HJB Equation) Proof. The proof follows from (Bellman’s Principle of Optimality), where we defineτ:=t+ ∆tas a small time step ∆t to get the following expanded expression for the value function: V t (x) = inf u E X u t:T ∼P u Z t+∆t t + Z T t+∆t 1 2 ∥u(X u s ,s)∥ 2 + c(X u s ,s) ds + Φ(X u T ) X u t = x = inf u E X u t:T ∼P u Z t+∆t t 1 2 ∥u(X u s ,s)∥ 2 + c(X u s ,s) ds + Z T t+∆t 1 2 ∥u(X u s ,s)∥ 2 + c(X u s ,s) ds + Φ(X T ) |z =:V t+∆t (X u t+∆t ) X u t = x = inf u E X u t:T ∼P u Z t+∆t t 1 2 ∥u(X u s ,s)∥ 2 + c(X u s ,s) ds + V t+∆t (X u t+∆t ) X u t = x = inf u 1 2 ∥u(X u t ,t)∥ 2 + c(X u t ,t) ∆t +O(∆t 2 ) + inf u E X u t:T ∼P u V t+∆t (X u t+∆t ) X u t = x (3.2) 63 Applying (Controlled Itˆo Formula withA u t ), the change in the value function over the interval [t,t + ∆t] expands into: V t+∆t (X u t+∆t )− V t (X u t ) = Z t+∆t t dV s (X u s ) | z Itˆo’s formula (2.7) = Z t+∆t t (∂ s +A u s )V s (X u s )ds + Z t+∆t 0 σ s ∇V s (X u s ) ⊤ dB s | z Itˆo integral vanishes under expectation (3.3) MovingV t (X u t ) to the right-hand side and applying the conditional expectationE[·|X u t =x] to both sides, we get: E[V t+∆t (X u t+∆t )|X u t = x] = E V t (X u t ) + Z t+∆t t (∂ s +A u s )V s (X u s )ds X t = x = V t (x) + (∂ t +A u t )V t (x)∆t + o(∆t)(3.4) Substituting this expression (3.4) into (3.2), we obtain the equation: V t (x) = inf u 1 2 ∥u(X u t ,t)∥ 2 + c(X u t ,t) ∆t + V t (x) + (∂ t +A u t )V t (x)∆t +O(∆t 2 )(3.5) Now, we can subtractV t (x) from both sides, divide by ∆tand take the continuous time limit ∆t→0 to get the expression: 0 = inf u 1 2 ∥u(x,t)∥ 2 + c(x,t) + (∂ t +A u t )V t (x) (3.6) Completing the square for the u-dependent terms, we get the form of the minimizer: inf u 1 2 ∥u(x,t)∥ 2 +⟨∇V t (x),σ t u(x,t)⟩ = inf u 1 2 ∥u(x,t)∥ 2 +⟨∇V t (x),σ t u(x,t)⟩+ σ 2 t 2 ∥∇V t (x)∥ 2 − σ 2 t 2 ∥∇V t (x)∥ 2 = inf u 1 2 ∥u(x,t) + σ t ∇V t (x)∥ 2 − σ 2 t 2 ∥∇V t (x)∥ 2 =⇒ u ⋆ (x,t) =−σ t ∇V t (x)(3.7) Plugging the expression for the optimalu ⋆ into (3.6) and rewriting the controlled generator in terms of the uncontrolled generator with (2.17), we recover the HJB equation: 0 = 1 2 ∥u ⋆ (x,t)∥ 2 + c(x,t) + (∂ t +A u ⋆ t )V t (x) 0 = 1 2 ∥u ⋆ (x,t)∥ 2 + c(x,t) + ∂ t V t (x) + (A t V t )(x) +⟨∇V t (x),σ t u ⋆ (x,t)⟩ 0 = σ 2 t 2 ∥∇V t (x)∥ 2 + c(x,t) + ∂ t V t (x) + (A t V t )(x)−σ 2 t ∥∇V t (x)∥ 2 ∂ t V t (x) =−(A t V t )(x) + σ 2 t 2 ∥∇V t (x)∥ 2 − c(x,t)(Hamilton-Jacobi-Bellman Equation) which is the HJB equation stated in (Value HJB Equation). Intuitively, this states that the time evolution of the value function∂ t V t (x) is equal to the instantaneous change in value induced by the uncontrolled dynamics−(A t V t )(x), the cost that arises from tilting the dynamics with the optimal control σ 2 t 2 ∥V t (x)∥ 2 , and the immediate running cost −c(x,t).□ Recall from Section 2.7 that the optimality conditions for (Dynamic SB Problem) with arbitrary reference dynamics are defined by the pair (ψ t ,p ⋆ t ), whereψ t is the Lagrange multiplier of the constrained optimization problem andp ⋆ t is the optimal marginal density evolution overt∈[0,T]. 64 Then, we showed that (ψ t ,p ⋆ t ) that characterize the dynamic SB problem satisfy a system of non- linear PDEs called the (HJB-FP System) which can then be transformed into a system of linear PDEs (Hopf-Cole PDEs) via the (Hopf-Cole Transform). Leveraging this idea, we can also write the SOC problem described by the (Value HJB Equation) into a linear PDE via the Hopf-Cole transform. Corollary 3.4 (Hopf-Cole Transform for Value Function). LetV t (x) be the value function that defines the solution to the (SOC Objective) and satisfies the (Value HJB Equation). Then, we define the change of variables: V t (x) =− logφ t (x) ⇐⇒ φ t (x) = e −V t (x) (3.8) where φ t (x) satisfies the linear PDE defined as: ∂ t φ t (x) +⟨f (x,t),∇φ t (x)⟩ + σ 2 t 2 ∆φ t (x)− c(x,t)φ t (x) = 0, φ T (x) = e −Φ(x) (Hopf-Cole Value PDE) Proof. First, we recall the (Value HJB Equation) given by: ∂ t V t (x) + (A t V t )(x)− σ 2 t 2 ∥∇V t (x)∥ 2 + c(x,t) = 0, V T (x) = Φ(x)(3.9) where (A t V t )(x) =⟨f (x,t),∇V t (x)⟩ + σ 2 t 2 ∆V t (x)(3.10) Following the analogous steps from Section 2.8, we express each term containing V t in terms of φ: (i) Time Derivative: ∂ t V t =−∂ t logφ t =− ∂ t φ t φ t . (i) Gradient Term: ∇V t =−∇ logφ t =− ∇φ t φ t =⇒ σ 2 t 2 ∥∇V t ∥ 2 = σ 2 t 2 ∥∇φ t ∥ 2 φ 2 t . (i) Generator Term: First, we expand the Laplacian term in (A t V t ) as: ∆V t =∇·∇(− logφ t ) =−∇· ∇φ t φ t =−∇· (φ −1 t ∇φ t ) =−∇· (φ −1 )·∇φ t + φ −1 t (∇·∇φ t ) | z product rule of divergences =− ∆φ t φ t + ∥∇φ t ∥ 2 φ 2 t (3.11) Then, the generator becomes: (A t V t ) = f,− ∇φ t φ t + σ 2 t 2 − ∆φ t φ t + ∥∇φ t ∥ 2 φ 2 t =− ⟨f,∇φ t ⟩ φ t − σ 2 t 2 ∆φ t φ t + σ 2 t 2 ∥∇φ t ∥ 2 φ 2 t (3.12) Substituting each term back into the HJB equation (3.9) and cancelling terms, we get: − ∂ t φ t φ t | z ∂ t V t (x) − ⟨f,∇φ t ⟩ φ t − σ 2 t 2 ∆φ t φ t + σ 2 t 2 ∥∇φ t ∥ 2 φ 2 t |z (A t V t )(x) − σ 2 t 2 ∥∇φ t ∥ 2 φ 2 t |z σ 2 t 2 ∥∇V t (x)∥ 2 +c(x,t) = 0 − ∂ t φ t φ t − ⟨f,∇φ t ⟩ φ t − σ 2 t 2 ∆φ t φ t + c(x,t) = 0 ∂ t φ t +⟨f,∇φ t ⟩ + σ 2 t 2 ∆φ t − c(x,t)φ t = 0(3.13) 65 where the final line follows from multiplying both sides by−φ t . Given the terminal condition V T (x) = Φ(x) defined in (3.9), we apply the change of variables to getφ T (x) =e −V T (x) =e −Φ(x) . Putting this together with (3.13), we have thatφ t satisfies the transformed linear PDE given by: ∂ t φ t (x) +⟨f (x,t),∇φ t (x)⟩ + σ 2 t 2 ∆φ t (x)− c(x,t)φ t (x) = 0, φ T (x) = e −Φ(x) (Hopf-Cole Value PDE) which concludes our derivation of the Hopf-Cole transform for the value function that solves (SOC Objective).□ It is worth noticing that the Hopf-Cole transform that we performed in Section 2.8 resulted in two linear PDEs that describe the evolution of the forward and backward Schr ̈odinger potentials (φ t , ˆφ t ), whereas we only derived a single linear PDE for the SOC problem. This is because while the (Dynamic SB Problem) has two terminal constraints at both marginalsp 0 =π 0 andp T =π T , the (SOC Objective) only has one terminal constraint for the value function atV T (x) = Φ(x), resulting in a single potential change-of-variablesφ t (x) =e −V t (x) that satisfies the backward (Hopf-Cole Value PDE) with terminal constraintφ T (x) =e −Φ(x) . Despite this difference in the two formulations, we make the following remark, which we explore further in Section 3.2. Remark 3.5. Although the Schr ̈odinger bridge problem imposes constraints on both the initial and terminal distributions, it can still be reformulated as an SOC problem. The key observation is that the initial distribution can be treated as the starting distribution of the controlled process, while the terminal constraint can be enforced through an appropriate terminal cost that penalizes deviations from the desired terminal marginal. We have shown that the value functionV t can be analyzed in the form of a linear PDE. We now present an alternative perspective by analyzing the evolution ofV t along stochastic trajectories generated under the uncontrolled reference measureQ. In particular, we show that the stochastic processX 0:T together with the value function evaluated along the trajectory satisfies a system of forward-backward stochastic differential equations (FBSDEs). In this formulation, the forward equation describes the evolution of the uncontrolled diffusion process, while the backward equation governs the dynamics of the value function along the stochastic path. Proposition 3.6 (Forward-Backward Stochastic Differential Equations). The optimal con- trolled process can be defined with a pair of forward and backward SDEs (FBSDEs) which define the evolution of the uncontrolled processX 0:T and the value functionV t (X t ), given by: dX t = f (X t ,t)dt + σ t dB t dV t (X t ) = σ 2 t 2 ∥∇V t (X t )∥ 2 − c(X t ,t) dt +∇V t (X u t ) ⊤ σ t dB t V T (X T ) = Φ(X T ) (3.14) In addition, for the controlled process (X u t ) t∈[0,T ] , the FBSDEs are given by: dX u t = (f (X u t ,t) + σ t u(X u t ,t))dt + σ t dB t dV t (X u t ) = σ 2 t 2 ∥∇V t (X u t )∥ 2 − c(X u t ,t) +⟨σ t u(X u t ,t),∇V t (X u t )⟩ dt +∇V t (X u t ) ⊤ σ t dB t V T (X u T ) = Φ(X u T ) (3.15) where g : R d → R is the terminal cost and σ t is the diffusion coefficient. 66 Proof. Applying (Uncontrolled Itˆo Formula with A t ) to the value function, we have: dV t (X t ) = ∂ t V t (X t )dt + (A t V t )(X t )dt +∇V t (X t ) ⊤ σ s dB t (3.16) From (Value HJB Equation), we can substitute∂ t V t (X t ) + (A t V t )(X t ) = σ 2 t 2 ∥∇V t (X t )∥ 2 − c(X t ,t) to get the backward evolution of the value function over the uncontrolled process as: dV t (X t ) = σ 2 t 2 ∥∇V t (X t )∥ 2 − c(X t ,t) dt +∇V t (X t ) ⊤ σ t dB t V T (X T ) = Φ(X T )(3.17) Applying (Controlled Itˆo Formula withA u t ) and the same steps as above to the value function evaluated along the controlled process X u 0:T , we get: dV t (X u t ) = ∂ t V t (X u t ) + (A u t V t )(X u t ) dt + σ t ∇V t (X u t ) ⊤ σ t dB t (Controlled Itˆo Formula with A u t ) 2.17 = ∂ t V t (X u t ) + (A t V t )(X u t ) +⟨σ t u(X u t ,t),∇V t (X u t )⟩ dt +∇V t (X u t ) ⊤ σ t dB t (3.18) Substituting the expression for ∂ t V t (X t ) + (A t V t )(X t ) from (Value HJB Equation), we have: dV t (X u t ) = σ 2 t 2 ∥∇V t (X u s )∥ 2 − c(X u t ,t) +⟨σ s u(X u t ,t),∇V t (X u t )⟩ ds +∇V t (X u t ) ⊤ σ t dB t (3.19) V T (X u T ) = Φ(X u T )(3.20) which recovers the backward evolution of the value function over the controlled process from the terminal constraint V T (X u T ) = Φ(X u T ).□ The FBSDE representation of the value function establishes the evolution of the value function under a stochastic process in the form of a backward SDE. By integrating this backward SDE along the trajectory and taking expectations, we can derive an explicit relationship between the cost functional J(x,t;u) for an arbitrary controluand the value functionV t (x) :=J ⋆ (x,t;u ⋆ ) defined with the optimal controlu ⋆ , which can be written with respect to the value function as shown in the following Proposition. Proposition 3.7 (Optimal Control and Value Function). Given a reference path measure Q, the relationship between the cost functional, or cost-to-go,J(x,t;u) and the value function V t (x) is defined as: J (x,t;u) = V t (x) + E X u t:T ∼P u Z T t 1 2 ∥σ s ∇V s (X u s ) +u(X u s ,s)∥ 2 X u t = x (Cost Functional) whereQis defined by the uncontrolled SDEdX t =f(X t ,t)dt+σ t dB t . Furthermore, the optimal control satisfies: u ⋆ (x,t) =−σ t ∇V t (x)(Optimal Control) Proof. First, we can write the value change over the time intervals ∈[t,T] using Itˆo’s lemma from Theorem 2.7 for the controlled stochastic process (X u t ) t∈[0,T ] and the value functionV t (x)∈ C 2,1 (R d × [0,T ]) to get the time derivative: dV s (X u s ) = ∂ s V s (X u s ) +⟨f (X u s ,s) + σ s u(X u s ,s),∇V s (X u s )⟩ + 1 2 σ 2 s ∆V s (X u s ) ds +∇V s (X u s ) ⊤ σ s dB s (3.21) 67 Then, we can derive the value difference over the interval s∈ [t,T ] as: V T (X u T )− V t (X u t ) = Z T t dV s (X u s ) = Z T t ∂ s V s (X u s ) +A s V s (X u s ) +⟨σ s u(X u s ,s),∇V s (X u s )⟩ ds + Z T t ∇V s (X u s ) ⊤ σ s dB s (3.22) We can substitute the expression for∂ s V s (X s ) + (A s V s )(X t ) from (Value HJB Equation) and complete the square to get: ∂ s V s (X u s )+A s V s (X u s ) +⟨σ s u(X u s ,s),∇V s (X u s )⟩ = σ 2 t 2 ∥∇V s (X u s )∥ 2 − c(X u s ,s) +⟨σ s u(X u s ,s),∇V s (X u s )⟩ = σ 2 t 2 ∥∇V s (X u s )∥ 2 +⟨σ s u(X u s ,s),∇V s (X u s )⟩ + 1 2 ∥u(X u s ,s)∥ 2 − 1 2 ∥u(X u s ,s)∥ 2 − c(X u s ,s) = 1 2 ∥σ s ∇V s (X u s ) +u(X u s ,s)∥ 2 − 1 2 ∥u(X u s ,s)∥ 2 − c(X u s ,s)(3.23) Substituting (3.23) back into (3.22) and settingV T (X u T ) = Φ(X u T ), we can rearrange and take the conditional expectation E[·|X u t = x] to get: Φ(X u T )− V t (X u t ) = Z T t 1 2 ∥σ s ∇V s (X u s ) +u(X u s ,s)∥ 2 − 1 2 ∥u(X u s ,s)∥ 2 − c(X u s ,s) ds + Z T t ∇V s (X u s ) ⊤ σ s dB s = Z T t 1 2 ∥σ s ∇V s (X u s ) +u(X u s ,s)∥ 2 ds− Z T t 1 2 ∥u(X u s ,s)∥ 2 − c(X u s ,s) ds |z (⋆) + Z T t ∇V s (X u s ) ⊤ σ s dB s (3.24) Observing that both Φ(X T ) and (⋆) appear in the conditional expectation of the cost functional J (x,t;u), we can rearrange and take expectations E[·|X t = x] to get: (⋆) + Φ(X u T ) = V t (X u t ) + Z T t 1 2 ∥σ s ∇V s (X u s ) +u(X u s ,s)∥ 2 + Z T t ∇V s (X u s ) ⊤ σ s dB s E[(⋆) + Φ(X u T ) X u t = x] | z =:J(x,t;u) = E V t (X u t ) + Z T t 1 2 ∥σ s ∇V s (X u s ) +u(X u s ,s)∥ 2 + Z T t ∇V s (X u s ) ⊤ σ s dB s | z vanishes under expectation X u t = x Recognizing that the left-hand side is the definition of the cost-to-go functionJ(x,t;u) and the expectation of the Itˆo integral vanishes, we conclude: J (x,t;u) = V t (x) + E " Z T t 1 2 ∥σ s ∇V s (X u s ) +u(X u s ,s)∥ 2 X u t = x # (Cost Functional) which establishes the relationship between the value function defined with the optimal controlu ⋆ and the cost-to-go under the arbitrary controlu. Since, the value function is equal to the optimal cost-to-go by definition, whereV t (x) :=J ⋆ (x,t;u ⋆ ), the exepectation must vanish at optimality. Therefore, we can derive the expression for the optimal control u ⋆ (x,t) from the condition: ∀(x,s)∈ R d × [0,T ], ∥σ s ∇V s (x) +u ⋆ (x,s)∥ 2 = 0 =⇒ u ⋆ (x,s) =−σ s ∇V s (x) (3.25) which recovers the expression in (Optimal Control).□ The value function can alternatively be expressed using expectations over an uncontrolled reference measureQ(Domingo i Enrich et al., 2024; Kappen et al., 2012). This representation, derived using the (Feynman-Kac Formula), will later allow us to connect stochastic optimal control with the Schr ̈odinger bridge formulation based on path measures. 68 Proposition 3.8 (Closed-Form Expression for Value Function (Equation 8 in Domingo i Enrich et al. (2024))). Given a reference path measureQ, the value functionV t (x) can be derived independently of the optimal control u ⋆ as: V t (x) =− log E X 0:T ∼Q " exp − Z T t c(X s ,s)ds + Φ(X T ) ! X t = x # (3.26) where Q is defined by the uncontrolled SDE dX t = f (X t ,t)dt + σ t dB t . Proof. This proof combines the (Value HJB Equation), (Hopf-Cole Value PDE), and the (Feynman- Kac Formula). First, we recall that the value function solves (Value HJB Equation), defined as: ∂ t V t +A t V t − 1 2 ∥σ ⊤ t ∇V t ∥ 2 + c = 0(3.27) which we show in Corollary 3.4 be transformed to a (Hopf-Cole Value PDE) via the change in variables φ t (x) = e −V t (x) defined as: ∂ t φ t (x) +⟨f (x,t),∇φ t (x)⟩ + σ 2 t 2 ∆φ t (x)− c(x,t)φ t (x) = 0, φ T (x) = e −Φ(x) (3.28) Applying the (Feynman-Kac Formula), we can writeφ t (x) as an expectation over uncontrolled stochastic processes as: φ t (x) = e −V t (x) = E X 0:T ∼Q exp − Z T t c(X s ,s)ds e −Φ(X T ) X t = x = E X 0:T ∼Q exp − Z T t c(X s ,s)ds− Φ(X T ) X t = x (3.29) Taking the logarithm on both sides and inverting the sign, we get: V t (x) =− log E X 0:T ∼Q exp − Z T t c(X s ,s)ds− Φ(X T ) X t = x (3.30) which defines the closed form expression for the value function as an expectation over uncontrolled stochastic processes.□ The Feynman-Kac representation above expresses the value function as a log-expectation with respect to the reference path measureQ. While this formulation provides a closed-form characterization of the optimal cost-to-go, it does not yet explicitly describe how the optimal controlled process differs from the reference dynamics. To make this relationship concrete, we interpret SOC as a change of measure on path space, where the optimal pathP ⋆ is obtained by reweighting the reference process Q using the change in the value function. Proposition 3.9 (Radon-Nikodym Derivative Between Optimal and Reference Path Measure). The Radon-Nikodym Derivative (RND) between the optimal controlled path measureP ⋆ and the reference path measure Q is defined as: dP ⋆ dQ (X 0:T ) = e −Φ(X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt (Optimal Path RND) which directly yields the optimal path measure: P ⋆ (X 0:T ) = 1 Z Q(X 0:T )e −Φ(X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt (Optimal Path Measure) Z := E Q h e −Φ(X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt i (Normalization Constant) 69 whereZis the normalization constant. Furthermore, given that the exponential factor depends only on the endpoints (X 0 ,X T ), the joint endpoint law of the optimal path measure P ⋆ (X 0 ,X T ) is given by: P ⋆ (X 0 ,X T ) = 1 Z Q(X 0 ,X T )e −Φ(X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt (3.31) Proof. To prove this, we leverage the form of the RND from Theorem 2.20 where the two path measures are generated via the SDEs: P ⋆ : dX t = (f (X t ,t) + σ t u ⋆ (X t ,t))dt + σ t dB t (3.32) Q : dX t = f (X t ,t)dt + σ t dB t (3.33) which yields the RND: dP ⋆ dQ (X 0:T ) = exp − 1 2 Z T 0 ∥u ⋆ (X t ,t)∥ 2 dt + Z T 0 u ⋆ (X t ,t) ⊤ dB t (3.34) Substituting the definition of the optimal control with the value functionu ⋆ (X t ,t) =−σ t ∇V t (X t ), we get: dP ⋆ dQ (X 0:T ) = exp − 1 2 Z T 0 ∥−σ t ∇V t (X t )∥ 2 dt + Z T 0 −σ t ∇V t (X t ) ⊤ dB t = exp − Z T 0 σ 2 t 2 ∥∇V t (X t )∥ 2 dt− Z T 0 ∇V t (X t ) ⊤ σ t dB t (3.35) Recall that (Value HJB Equation) also contains the expression σ 2 t 2 ∥∇V t (X t )∥ 2 which can be isolated as: ∂ t V t (x) =−(A t V t )(x) + σ 2 t 2 ∥∇V t (x)∥ 2 − c(x,t) =⇒ σ 2 t 2 ∥∇V t (x)∥ 2 = ∂ t V t (x) + (A t V t )(x) + c(x,t) (3.36) Substituting this into the integral, we have: dP ⋆ dQ (X 0:T ) = exp − Z T 0 (∂ t V t (X t ) + (A t V t )(X t ) + c(X t ,t))dt− Z T 0 ∇V t (X t ) ⊤ σ t dB t = exp − Z T 0 (∂ t V t (X t ) + (A t V t )(X t ))dt−∇V t (X t ) ⊤ σ t dB t | z dV t (X t ) − Z T 0 c(X t ,t)dt (3.37) Since integrating the Itˆo integral of the value process is equal to the value difference, we can write: dP ⋆ dQ (X 0:T ) = exp − Z T 0 dV t (X t ) |z (V T (X T )−V 0 (X 0 )) − Z T 0 c(x,t)dt = exp − V T (X T ) + V 0 (X 0 )− Z T 0 c(x,t)dt (3.38) Multiplying both sides by the reference path measure and scaling by a normalization factor, we get the expression for the optimal path measure as: P ⋆ (X 0:T ) = 1 Z e −V T (X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt Q(X 0:T ) (3.39) 70 where Z is defined such that the distribution integrates to one: Z 1 Z e −V T (X T )+V 0 (X 0 ) Q(X 0:T )− Z Z T 0 c(x,t)dtQ(X 0:T ) = 1 Z E Q h e −V T (X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt i = 1 =⇒ Z := E Q h e −V T (X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt i (3.40) which concludes our derivation of the expression for the optimal RND and optimal path measure. Since the exponential scaling factor depends only on the endpoints (X 0 ,X T ), we can derive the expression for the endpoint law: P ⋆ (X 0 ,X T ) = 1 Z Q(X 0 ,X T )e −V T (X T )+V 0 (X 0 )− R T 0 c(X t ,t)dt (3.41) which means that the endpoint law of the optimal path measure is an exponential tilting of the reference path measure by the initial and terminal value functions. Substituting the terminal constraint V T (X T ) = Φ(X T ) yields the final expressions in Proposition 3.9.□ The expression for (Optimal Path Measure) reveals the key challenge in solving the SOC problem. That is, the optimal path measure is a tilted version of the reference path measure by not only the terminal cost−V T (X T ) but also the initial value functionV 0 (X 0 ), which is defined as an expectation over stochastic paths as defined in Proposition 3.8 and is intractable. This motivates the definition of memoryless reference processes, which removes the dependency of the optimal path on the initial value function. Memoryless Reference Process Eliminates Initial Value Function Bias Consider a reference processQwhere the joint distribution of initial and terminal states is independent, such that the endpoint distribution factorizes: Q(X 0 ,X T ) = q 0 (X 0 )q T (X T )(3.42) From Proposition 3.9, we show that the endpoint distribution of the optimal path measureP ⋆ is an exponential tilting of the reference law, given by: P ⋆ (X 0 ,X T ) = 1 Z Q(X 0 ,X T )e −Φ(X T )+V 0 (X 0 ) (3.43) where we consider the common case with no running cost functionc≡0. Then, the terminal distribution under the optimal process P ⋆ satisfies: p ⋆ T (X T ) = Z R d P ⋆ (X 0 ,X T )dX 0 = Z R d Q(X 0 ,X T )e −Φ(X T )+V 0 (X 0 ) dX 0 = Z R d q 0 (X 0 )q T (X T )e −Φ(X T ) e V 0 (X 0 ) dX 0 = q T (X T )e −Φ(X T ) Z R d q 0 (X 0 )e V 0 (X 0 ) dX 0 | z constant normalization = 1 Z q T (X T )e −Φ(X T ) (3.44) where the initial value function integrates to a constant normalization factor, eliminating the need to computeV 0 (X 0 ) explicitly. Furthermore, we can define Φ(X T ) such that the terminal marginal is exactly the target distribution π T (X T ) as: 1 Z q T (X T )e −Φ(X T ) = π T (X T ) =⇒ q T (X T ) Zπ T (X T ) = e Φ(X T ) =⇒ Φ(X T ) = log q T (X T ) π T (X T ) − logZ(3.45) 71 which yields a tractable objective that does not require computingV 0 (X 0 ). Examples of memoryless reference processes include: (i)Linear reference driftf:=A t X t +b t , for someA t ∈ R d×d andb t ∈ R d , and noise that grows with time such that the initial distribution is a standard zero-mean Gaussian X 0 ∼N(0,I d ). This includes the variance-preserving SDE used in Song et al. (2020b). Q : dX t =− 1 2 β t X t dt + p β t dB t , X 0 ∼N (0,I d ) =⇒ Q(X 0 ,X T )≈ q 0 (X 0 )q T (X T ) (i)Brownian motion reference driftf:= 0 and a initial distribution that is a Dirac delta π 0 := δ 0 . Q : dX t = σ t dB t , X 0 = 0 =⇒ Q(X 0 ,X T )≈ δ 0 (X 0 )q T (X T ) While this shows that we can solve the SOC problem without computing the initial value function V 0 (X 0 ) by explicitly designing the reference process to be memoryless, this construction restricts us to a limited class of relatively uninformative reference dynamics, which may not provide a meaningful prior for a given system. To overcome this limitation, we will reformulate the SOC objective by absorbing the initial value function into the terminal cost, which allows for a broader class of reference processes and more expressive prior dynamics. 3.2 Schr ̈odinger Bridges with Stochastic Optimal Control In this section, we will adapt stochastic optimal control (SOC) theory for solving the (Dynamic SB Problem) with arbitrary prior dynamics and initial distributions. First, we recall that the Hopf-Cole transform from Theorem 2.24 expresses the optimal controlu ⋆ and probability densityp ⋆ t using a pair of forward-backward SB potentials (φ t , ˆφ t ) as: u ⋆ (x,t) = σ t ∇ logφ t (x), p ⋆ t (x) = φ t (x) ˆφ t (x)(3.46) where (φ t , ˆφ t ) are defined in (Schr ̈odinger Potentials) as: ( φ t (x) = R R d Q T|t (y|x)φ T (y)dy ˆφ t (x) = R R d Q t|0 (x|y) ˆφ T (y)dy s.t. ( π 0 (x) = φ 0 (x) ˆφ 0 (x) π T (x) = φ T (x) ˆφ T (x) (Schr ̈odinger Potentials) Now, we can rewrite the original SOC problem in (3.1) by observing that the optimal control can be written in two equivalent ways: using the value functionV t (x) (Proposition 3.7) or the Schr ̈odinger potentialφ t (x) (3.46). This allows us to derive the relationship between the value function and the SB potential as: u ⋆ (x,t) =−σ t ∇V t (x) = σ t ∇ logφ t (x) =⇒ V t (x) =− logφ t (x)(SB Value Function) Since the terminal cost is given by the terminal value function Φ(x) =V T (x), we can also write the terminal cost in terms of the SB potentials as: Φ(x) = V T (x) =− logφ T (x) =− log π T (x) ˆφ T (x) = log ˆφ T (x) π T (x) (SB Terminal Cost) where we substituteπ T (x) =φ T (x)ˆφ(x) =⇒ φ T (x) = π T (x) ˆφ t (x) from (Schr ̈odinger Potentials). Substituting (SB Value Function) and (SB Terminal Cost) into the results from Proposition 3.9, we can write the optimal path RND and path measure P ⋆ that solves the dynamic SB problem as: dP ⋆ dQ (X 0:T ) = exp (−Φ(X T ) + V 0 (X 0 )) = exp − log ˆφ T (X T ) π T (X T ) − logφ 0 (X 0 ) (Optimal SB Path RND) P ⋆ (X 0:T ) = 1 Z Q(X 0:T ) exp − log ˆφ T (X T ) π T (X T ) − logφ 0 (X 0 ) (Optimal SB Path Measure) 72 Using (SB Value Function) and (SB Terminal Cost), we also define the alternative Schr ̈odinger bridge form of the SOC problem which yields the optimal SB solution. Definition 3.10 (Schr ̈odinger Bridge with Stochastic Optimal Control (SB-SOC)). Given the Schr ̈odinger potentials (φ t , ˆφ t ) that satisfy the linear PDEs in (Hopf-Cole PDEs), the (SOC Objective) can be expressed in terms of (φ t , ˆφ t ) as: inf u E X u 0:T ∼P u " Z T 0 1 2 ∥u(X u t ,t)∥ 2 dt + log ˆφ T (X u T ) π T (X u T ) # (SB-SOC Objective) s.t. dX u t = (f (X u t ,t) + σ t u(X u t ,t))dt + σ t dB t , X u 0 ∼ π 0 whereu(x,t) is the control drift which produces the path measureP u ,f(x,t) is the drift of the reference processQ,σ t is the diffusion coefficient, anddB t isd-dimensional Brownian motion. Given the (SB-SOC Objective), we can define an cost-to-go analogously to (Cost Functional) from any fixed point (x,t)∈ R d × [0,T ] at time t under the control u as: J (x,t;u) := E X u t:T ∼P u " Z T t 1 2 ∥u(X u s ,s)∥ 2 ds + log ˆφ T (X u T ) π T (X u T ) X u t = x # (3.47) We can also show that (SB-SOC Objective) eliminates the initial value bias and does not require defining a memoryless reference process as discussed in Box 3.1. Proposition 3.11 (Schr ̈odinger Potential Eliminates the Initial Value Bias). The (SB-SOC Objective) does not require computing the initial value functionV 0 (x). In particular, under the optimal controlu ⋆ , the induced path measureP ⋆ automatically satisfies the terminal marginal constraint p ⋆ T (x T ) = π T (x T ) independently of the initial value function V 0 (x). Proof. Starting from the SB-SOC representation of the (Optimal SB Path Measure), we can write the optimal endpoint law as: P ⋆ (X 0 ,X T ) = 1 Z Q(X 0 ,X T ) exp − log ˆφ T (X T ) π T (X T ) − logφ 0 (X 0 ) (3.48) Integrating overX 0 , we have that the terminal marginal of the optimal controlled pathp ⋆ T is equal to: p ⋆ T (X T ) = Z R d Q(X 0 ,X T ) exp − log ˆφ T (X T ) π T (X T ) − logφ 0 (X 0 ) dX 0 (3.49) Rearranging the exponential term, we have: p ⋆ T (X T ) = π T (X T ) ˆφ T (X T ) Z R d Q(X 0 ,X T ) 1 φ 0 (X 0 ) dX 0 (3.50) Using the factorizationQ(X 0 ,X T ) =Q(X T |X 0 )π 0 (X 0 ) and the boundary conditionπ 0 (x) = φ 0 (x) ˆφ 0 (x), we get: as follows: p ⋆ T (X T ) = π T (X T ) ˆφ T (X T ) Z R d Q(X T |X 0 )π 0 (X 0 ) ˆφ 0 (X 0 ) π 0 (X 0 ) dX 0 = π T (X T ) = π T (X T ) ˆφ T (X T ) Z R d Q(X T |X 0 ) ˆφ 0 (X 0 )dX 0 | z = ˆφ T (X T ) = π T (X T )(3.51) 73 which shows that the optimal path measureP ⋆ under the optimal controlu ⋆ that solves (SB-SOC Objective) exactly matches the terminal distributionp ⋆ T =π T , without dependence on the initial value V 0 (X 0 ).□ We can now characterize the full optimal path measure induced by the optimal controlu ⋆ . In particular, we will show that the Schr ̈odinger potentials (φ t , ˆφ t ) provide a clean factorization of the optimal bridge measure with respect to the reference processQ, which reveals how the endpoint potentials reweight trajectories of the reference dynamics to produce the Schr ̈odinger bridge. Furthermore, the optimal path densityP ⋆ (X 0:T ), as well as its associated marginal densityp ⋆ t and joint density p ⋆ s,t (y,x), can be written explicitly in terms of the potentials as stated below. Proposition 3.12 (Optimal Path Density of SB-SOC). The Schr ̈odinger bridge path measure that solves (SB-SOC Objective) can be written with respect to the SB potentials as: P ⋆ (X 0:T ) = 1 Z Q(X 0:T )φ T (X T ) ˆφ 0 (X 0 ) π 0 (X 0 ) = 1 Z Q(X 0:T |X 0 )φ T (X T ) ˆφ 0 (X 0 )(SB-SOC Path) and the marginal density at time t can be factorized as: p ⋆ t (x) = ˆφ t (x)φ t (x)(SB-SOC Density) Additionally, for any s≤ t, the joint density of X s = y and X t = x satisfies: p ⋆ s,t (y,x) = Q(X t = x|X s = y) ˆφ s (y)φ t (x), s≤ t(SB-SOC Joint Density) Proof. Using the alternative definition of the value function, we can also derive the optimal SB path measureP ⋆ with respect to the Schr ̈odinger potentials. From Proposition 3.9, we showed that the optimal path measure can be written as: P ⋆ (X 0:T ) = 1 Z Q(X 0:T ) exp −V T (X T ) + V 0 (X 0 )− Z T 0 c(X t ,t)dt (3.52) Since the dynamic SB problem is defined as a KL minimization problem, which only produces a quadratic control cost, we havec≡0. Therefore, substituting the (SB Terminal Cost) and (SB Value Function), we get: P ⋆ (X 0:T ) = 1 Z Q(X 0:T ) exp (−V T (X T ) + V 0 (X 0 )) = 1 Z Q(X 0:T ) exp (logφ T (X T )− logφ 0 (X 0 )) = 1 Z Q(X 0:T )φ T (X T ) 1 φ 0 (X 0 ) = 1 Z Q(X 0:T )φ T (X T ) ˆφ 0 (X 0 ) π 0 (X 0 ) = 1 Z Q(X 0:T ) π 0 (X 0 ) |z =Q(X 0:T |X 0 ) φ T (X T ) ˆφ 0 (X 0 ) = 1 Z Q(X 0:T |X 0 )φ T (X T ) ˆφ 0 (X 0 )(3.53) where we use the identityQ(X 0:T ) =Q(X 0:T |X 0 )π 0 (X 0 ). To obtain the marginal density at time tgiven byp ⋆ t we can define the distribution over pathsX 0:T whereX t =xand integrate over all paths X 0:T to get: p ⋆ t (x) = Z δ(X t −x)φ T (X T ) ˆφ 0 (X 0 )Q(dX 0:T |X 0 )(3.54) Since the integrand is multiplied only by the potentials evaluated atX 0 andX T , we integrate out the path coordinates leaving onlyX 0 andX T and apply the Markov property of the reference path 74 measure to get a simplified expression for the marginal density: p ⋆ t (x) = Z Q(X T ,X t = x|X 0 )φ T (X T ) ˆφ 0 (X 0 )dX 0 dX T = Z Z Q(X T |X t = x)Q(X t = x|X 0 )φ T (X T ) ˆφ 0 (X 0 )dX 0 dX T = Z Q(X T |X t = x)φ T (X T )dX T | z = ˆφ t (x) Z Q(X t = x|X 0 ) ˆφ 0 (X 0 )dX 0 |z =φ t (x) = ˆφ t (x)φ t (x)(3.55) To derive the joint distribution for arbitrarys≤ t, we can include additional conditioning onX s =y and apply the Markov property of Q similarly to (SB-SOC Density) to get: p ⋆ s,t (y,x) = Z Q(X T ,X t = x,X s = y|X 0 )φ T (X T ) ˆφ 0 (X 0 )dX 0 dX T = Z Z Q(X T |X t = x)Q(X t = x|X s = y)Q(X s = y|X 0 )φ T (X T ) ˆφ 0 (X 0 )dX 0 dX T = Q(X t = x|X s = y) Z Q(X T |X t = x)φ T (X T )dX T | z = ˆφ t (x) Z Q(X t = x|X 0 ) ˆφ 0 (X 0 )dX 0 |z =φ t (x) = Q(X t = x|X s = y) ˆφ t (x)φ t (x)(3.56) which is the joint probability density ofX s =yat timesandX t =xat timetunder the optimal SB-SOC path measure.□ Together, these results show that the SOC formulation of the Schr ̈odinger bridge can be fully characterized by the reference dynamicsQand the terminal Schr ̈odinger potentialsφ 0 andˆφ T , with the intermediate potentials defined via the (Schr ̈odinger Potentials) without the need to explicitly compute the value function. Building on this reformulation, we can now derive tractable training objectives that estimate the optimal control by sampling stochastic paths from proposal SDEs and minimizing divergences between control drift with respect to the reference drift. 3.3 Objectives for Solving the SOC Problem In this section, we introduce three different objective functions that can be used to solve the SOC problem defined in (SOC Objective): the relative-entropy loss L RE , the log-variance loss L LV , and the cross-entropy loss L CE (N ̈usken et al., 2021). Since the SOC objective is inherently a KL divergence between the controlled path measureP u generated withuand the optimal path measureP ⋆ :=P u ⋆ generated with the optimal controlu ⋆ , it is natural to consider a loss function that minimizes the KL divergence or the reverse KL divergence between P u and P ⋆ defined as: KL(P u ∥P ⋆ ) = E P u log dP u dP ⋆ ,KL(P ⋆ ∥P u ) = E P ⋆ log dP ⋆ dP u (3.57) which both yield a unique minimizer atP u =P ⋆ whenKL(P u ∥P ⋆ ) =KL(P ⋆ ∥P u ) = 0. These two divergences are exactly what define the relative-entropy (RE) and cross-entropy (CE) losses (N ̈usken et al., 2021). In the following definitions, we will define both its path measure form and derive its path-integral form with respect to the control drifts u. 75 Definition 3.13 (Relative Entropy (RE) Loss). The relative entropy (RE) loss between the controlled path measureP u and the optimal path measureP ⋆ is defined as the KL divergence: L RE (P u , P ⋆ ) := KL(P u ∥P ⋆ ) = E P u log dP u dP ⋆ (Relative Entropy Objective) Letu(x,t) denote the control that generatesP u and letX u 0:T = (X u t ) t∈[0,T ] denote a stochastic process under the (Controlled SDE). Then, the RE loss takes the path-integral form: L RE (u) := E X u 0:T ∼P u 1 2 Z T 0 ∥u(X u t ,t)∥ 2 dt + Φ(X u T ) + Z T 0 c(X u t ,t)dt (RE Loss) Derivation. To derive this loss, we leverage (Girsanov’s Theorem) and our derivation for the Radon- Nikodym derivative between the optimal and reference path measure derived in Proposition 3.12. Since P u is the controlled version of the reference process Q, we write: E P u log dP u dP ⋆ = E P u log dP u dQ + log dQ dP ⋆ = E P u log dP u dQ |z Girsanov − log dP ⋆ dQ (3.58) SinceP u andQdiffer only by the control driftσu(x,t), we can apply (Path RND) to get write the first term dP u dQ as: dP u dQ (X 0:T ) = exp − 1 2 Z T 0 ∥u(X t ,t)∥ 2 dt + Z T 0 u(X t ,t) ⊤ dB t (3.59) Substituting this and (Optimal Path RND), we get the integral form of the relative-entropy objective L RE as: L RE (u) = E X u 0:T ∼P u log dP u dQ (X u 0:T )− log dP ⋆ dQ (X u 0:T ) = E X u 0:T ∼P u − 1 2 Z T 0 ∥u(X u t ,t)∥ 2 dt + Z T 0 u(X u t ,t) ⊤ dB t + V T (X u T )− V 0 (X u 0 ) + Z T 0 c(X u t ,t)dt (3.60) By definition, we haveV T (X u T ) = Φ(X u T ) and the initial distribution is fixed atX 0 ∼ π 0 soV 0 (X 0 ) is constant. Then, we get the final form of L RE (u) as: L RE (u) := E X u 0:T ∼P u 1 2 Z T 0 ∥u(X u t ,t)∥ 2 dt + Φ(X u T ) + Z T 0 c(X u t ,t)dt (3.61) which is exactly (RE Loss) in our definition.□ While the relative-entropy loss has a unique minimizer whenP u =P ⋆ , optimizinguusing the gradient ∇ u L RE (u) with respect touor its parameters∇ θ L RE (u θ ), requires differentiating through the full SDE trajectories (X u t ) t∈[0,T ] due to the expectation overP u . In practice, if (X u t ) t∈[0,T ] is generated via an SDE solver, like the Euler-Maruyama method, this requires storing the full computational graph at each simulation step, which is memory-intensive. To obtain a practical estimator, the REINFORCE trick allows us to rewrite (Relative Entropy Objective) as an expectation under a stop-gradient sampling measureP ̄u , whereQ ̄u :=stopgrad(Q u ) is the non-gradient-tracking controlled generator (Mnih et al., 2014; Ranganath et al., 2014; Williams, 1992). 76 Definition 3.14 (REINFORCE Relative Entropy (RERF) Loss). The REINFORCE rela- tive entropy (RERF) loss is defined as an expectation overP ̄u , whereQ ̄u :=stopgrad(Q u ) is the non-gradient-tracking controlled generator: L RERF (P u , P ⋆ ) := E P ̄u log dP u dP ̄u log dP ⋆ dP ̄u + C (RERF Loss) whereC ∈ Ris any constant. Crucially, the gradient aligns with the gradient of the relative- entropy loss, i.e., ∇ u KL(P u ∥P u ) =∇ u L RERF (P u , P ⋆ ). While the (RERF Loss) provides an unbiased estimator of the relative-entropy gradient that satisfies ∇ u KL(P u ∥P u ) =∇ u L RERF (P u , P ⋆ ), it is important to note that it should be interpreted as a computational surrogate rather than a true loss function, as decreasing its value does not necessarily correspond to a monotonic reduction of the KL divergence itself. Instead, its role is to provide a tractable estimator of the gradient needed to learn the optimal control that induces the target path measure. An alternative objective that is a true loss function and doesn’t require an expectation overP u is the cross-entropy (CE) loss, which is simply the reverse KL divergence, where the expectation is over the fixed optimal path measure P ⋆ instead of P u . Definition 3.15 (Cross Entropy (CE) Loss). The cross entropy (CE) loss between the controlled path measureP u and the optimal path measureP ⋆ is defined as the KL divergence: L CE (P u , P ⋆ ) := KL(P ⋆ ∥P u ) = E P ⋆ log dP ⋆ dP u (Cross Entropy Objective) Letv(x,t) denote an arbitrary fixed control drift that generatesP v and letX v 0:T = (X v t ) t∈[0,T ] denote the stochastic process under P v . Then, the CE loss takes the path-integral form: L CE (u) = 1 Z E X v 0:T ∼P v exp −g(X v T )− Z T 0 c(X v t ,t)dt− 1 2 Z T 0 ∥v(X v t ,t)∥ 2 dt− Z T 0 v(X v t ,t) ⊤ dB v t 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 (u·v)(X v t ,t)dt− Z T 0 u(X v t ,t) ⊤ dB v t − g(X v T )− Z T 0 c(X v t ,t)dt + C (CE Loss) Derivation. This proof follows similar steps to the relative-entropy E P ⋆ log dP ⋆ dP u = E P v dP ⋆ dP v log dP ⋆ dP u = E P v dP ⋆ dP v log dP ⋆ dQ dQ dP u = E P v dP ⋆ dQ dQ dP v | z (⋆) log dP ⋆ dQ − log dP u dQ |z (⋄) (3.62) Applying Girsanov’s theorem and substituting (Optimal Path RND) into (⋄), we get the integral form of the relative-entropy objective L RE as: log dP ⋆ dQ (X v 0:T )− log dP u dQ (X v 0:T ) =−V T (X v T ) + V 0 (X v 0 )− Z T 0 c(X v t ,t)dt+ 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 u(X u t ,t) ⊤ dB Q t = 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 u(X v t ,t) ⊤ dB t − g(X v T )− Z T 0 c(X v t ,t)dt + C | z (⋄) (3.63) 77 Since the Brownian motion of the RND is under the reference path measureQin the denominator, we need to transform it to match the path measure under whichX v 0:T is generated. By (Girsanov’s Theorem), we substitute dB Q t = dB v t +v(X t ,t)dt which gives us: − V T (X v T ) + V 0 (X v 0 )− Z T 0 c(X v t ,t)dt + 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 u(X v t ,t) ⊤ (dB v t +v(X v t ,t)dt) = 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 (u·v)(X v t ,t)dt− Z T 0 u(X v t ,t) ⊤ dB v t − g(X v T )− Z T 0 c(X v t ,t)dt + C |z (⋄) (3.64) where we setV T (X v T ) =g(X v T ) and denoteV 0 (X v 0 ) as a constant. For (⋆), we can derive the (Path RND) ofQwith respect toP v , where the difference between the control drifts is−v(X v t ,t), and substitute (Optimal Path RND) to get: dP ⋆ dQ (X v 0:T ) dQ dP v (X v 0:T ) = 1 Z exp −V T (X v T ) + V 0 (X v 0 )− Z T 0 c(X v t ,t)dt exp − 1 2 Z T 0 ∥v(X v t ,t)∥ 2 dt− Z T 0 v(X v t ,t) ⊤ dB v t = 1 Z exp −g(X v T )− Z T 0 c(X v t ,t)dt− 1 2 Z T 0 ∥v(X v t ,t)∥ 2 dt− Z T 0 v(X v t ,t) ⊤ dB v t | z (⋆) (3.65) Substituting the expanded forms for (⋆) and (⋄), we get: L CE (u) = 1 Z E X v 0:T ∼P v exp −g(X v T )− Z T 0 c(X v t ,t)dt− 1 2 Z T 0 ∥v(X v t ,t)∥ 2 dt− Z T 0 v(X v t ,t) ⊤ dB v t |z (⋆) 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 (u·v)(X v t ,t)dt− Z T 0 u(X v t ,t) ⊤ dB v t − g(X v T )− Z T 0 c(X v t ,t)dt | z (⋄) + C (3.66) which recovers the (CE Loss) from our definition.□ Since the pathsX v 0:T are generated under the fixed controlled processP v instead of the path that is being optimizedP u , it is considered an off-policy objective. This admits a much more computationally tractable objective as the gradient∇ u L CE (u) no longer depends on the SDE trajectories, and computing the objective does not require differentiating through or maintaining the computational graph of the SDE solver. Typically, the off-policy controlvis defined as the non-gradient-tracked control being optimizedv:=stopgrad(u), which generates paths from the controlled SDE without maintaining the computational graph used to generate it. We also note that while bothL RE andL CE are uniquely minimized atP u =P ⋆ , theL CE is convex inP u , which yields a more favorable optimization landscape. Remark 3.16 (Convexity of the Cross-Entropy Objective). The cross-entropy lossL CE (P u , P ⋆ ) = KL(P ⋆ ∥P u ) is convex in the controlled path measureP u . This follows from the fact that P u 7→ log P u is convex when the reference distributionP ⋆ is fixed. Therefore, minimizing the CE objective corresponds to a convex optimization problem in the space of path measures, and the global minimum is achieved when P u = P ⋆ . Now that we have defined two objectives corresponding to the KL divergence between path measures, we consider an alternative class of objectives that aims to minimize the variance between the controlled and optimal path measures. To build the intuition behind the variance-based losses, we 78 recall that a stochastic processX u 0:T ∼ P u generated under a controlled process can be reweighted by the Radon-Nikodym derivative such that the law matches the optimal path measure P ⋆ : P ⋆ (X u 0:T ) | z optimal measure = dP ⋆ dP u (X u 0:T ) | z importance weight P u (X u 0:T ) | z paths under u (3.67) However, whenP u deviates far from the optimal measureP ⋆ , the importance weights will vary significantly for different pathsX u 0:T , with some weights being huge and others small. This means that the variance of the importance weight can be a measure of the similarity betweenP u andP ⋆ . Crucially, the variance is only zero when the importance weight is a constant that is not dependent on the pathX u 0:T . Since the RND is constant only when the two measures are equalP u =P ⋆ up to a normalization, we have: Var X u 0:T ∼P u dP ⋆ dP u (X u 0:T ) = 0 ⇐⇒ dP ⋆ dP u (X u 0:T ) = const. ⇐⇒ P u = P ⋆ (3.68) Similar to the cross-entropy objective, we want to prevent taking gradients under paths generated directly from the controlled path measure that we are optimizingP u , so we define an arbitrary, fixed controlvand sample pathsX v 0:T ∼ P v . Using these paths off-policy paths, the importance weights are still minimized when the RND is constant. Using this idea, we define the variance loss and log-variance loss. Definition 3.17 (Variance and Log-Variance Losses). The variance and log-variance (LV) loss between the controlled path measureP u and the optimal path measureP ⋆ is defined as the KL divergence: L Var P v (P u , P ⋆ ) := Var P v dP ⋆ dP u , L log Var (P u , P ⋆ ) := Var P v log dP ⋆ dP u (3.69) Letu(x,t) denote the control being optimized andv(x,t) denote an arbitrary fixed control that generates the stochastic paths (X v t ) t∈[0,T ] . Then, the variance and log-variance losses take the path-integral form: L Var P v (u) := 1 Z 2 Var P v e F u,v −Φ(X v T ) , L log Var (u) := Var P v (F u,v − Φ(X v T ))(Variance Losses) where F u,v is given by the expression: F u,v = 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 (u·v)(X v t ,t)dt− Z T 0 u(X v t ,t) ⊤ dB t − Z T 0 c(X v t ,t)dt(3.70) Derivation. We can break down the variance losses as follows: Var P v dP ⋆ dP u = Var P v dP ⋆ dQ dQ dP u ,Var P v log dP ⋆ dP u = Var P v log dP ⋆ dQ − log dP u dQ (3.71) 79 Expanding these terms similar form as (3.63), where dB Q t = dB v t +v(X v t ,t)dt, we get: Var X v 0:T ∼P v dP ⋆ dP u (X v 0:T ) = Var X v 0:T ∼P v dP ⋆ dQ dQ dP u (X v 0:T ) = 1 Z 2 Var X v 0:T ∼P v exp 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 (u·v)(X v t ,t)dt− Z T 0 u(X v t ,t) ⊤ dB v t | z =:F u,v −g(X v T )− Z T 0 c(X v t ,t)dt Var X v 0:T ∼P v log dP ⋆ dP u (X v 0:T ) = Var X v 0:T ∼P v log dP ⋆ dQ (X v 0:T )− log dP u dQ (X v 0:T ) = Var X v 0:T ∼P v 1 2 Z T 0 ∥u(X v t ,t)∥ 2 dt− Z T 0 (u·v)(X v t ,t)dt− Z T 0 u(X v t ,t) ⊤ dB v t |z =:F u,v − g(X v T )− Z T 0 c(X v t ,t)dt + C Then, we can defineF u,v := 1 2 R T 0 ∥u(X v t ,t)∥ 2 dt− R T 0 (u·v)(X v t ,t)dt− R T 0 u(X v t ,t) ⊤ dB v t and rewrite the variance losses as: L Var P v (u) := 1 Z 2 Var P v e F u,v −Φ(X v T ) , L log Var P v (u) := Var P v (F u,v − Φ(X v T ))(3.72) which recovers the (Variance Losses) from our definition.□ Comparing the (CE Loss) and (Variance Losses), we observe that they both sample paths from an arbitrary fixed path measureP v with controlv. However, the (CE Loss) does not change with different choices ofv, since the dependence onP v cancels out when scaling with the RND dP ⋆ /dP v . In contrast, the variance-based objectives change with respect to different definitions ofv, since the distribution under which the variance is calculated P v , changes with different v. Remark 3.18 (Variance Loss Depends on Sampling Distribution). The variance-based objectives in (Variance Losses) depend on the definition ofvused to sampleX v 0:T ∼ P v . However, it holds that the variance is minimized exactly when the controlled measureP u matches the optimal measure P ⋆ : ∀u∈U, L Var P v (u) = 0 ⇐⇒ u = u ⋆ and L log Var P v (u) = 0 ⇐⇒ u = u ⋆ (3.73) This is easy to see given that the RND between the optimal path measure and itself must be equal to one, i.e., dP ⋆ /dP ⋆ ≡1, regardless of the path on which it is evaluated, and the variance of a constant is zero. We have introduced three variations of the SOC objective, which are all derived from the RND between path measures. The key insight is that all three objectives can be estimated from paths generated from a tractable SDE, enabling practical learning of the optimal bridge dynamics. Although the objectives differ in their sampling laws, training stability, and convergence guarantees, they are all minimized when the controlled dynamics recover the optimal bridge P u = P ⋆ . 3.4 Closing Remarks for Section 3 In this section, we introduced the stochastic optimal control (SOC) problem, which defines a path- space variational objective that minimizes a running cost and terminal cost function generated by a controlled SDE. Leveraging dynamic programming theory, we introduce the value functionV t , which solves the Hamilton-Jacobi-Bellman equation similarly to the Lagrangianψ t defined in Section 2.7 and connect it to the optimal control driftu ⋆ and optimal path measure Radon-Nikodym derivative. A key observation is that the quadratic control cost in the SOC objective is equivalent to the KL divergence between the controlled and reference Itˆo processes from Corollary 2.21. This connection allows us to reformulate the dynamic Schr ̈odinger bridge problem as an SOC objective in which the 80 terminal cost is expressed as a log-ratio involving the backward Schr ̈odinger potentialˆφ T and the terminal marginal constraint π T . This reformulation avoids the need for explicit couplings between π 0 and π T , and we show that the backward potential ˆφ T effectively absorbs the dependence on the initial valueV 0 (X 0 ) appearing in classical SOC formulations, making the framework applicable to arbitrary initial distributions and reference dynamics. Building on this formulation, we introduced several tractable training objectives that are uniquely minimized by the optimal control driftu ⋆ and its corresponding path measureP ⋆ . A key step in deriving these objectives is using Girsanov’s theorem to express the Radon–Nikodym derivative between the controlled and reference path measures directly in terms of the control drift, yielding closed-form expressions and a practical way to evaluate and optimize path-space divergences using trajectories simulated from the controlled SDE. Now that we have explored several complementary formulations of the SB problem, from the static formulation to dynamic formulations with connections to SOC, we next turn to the practical construction of Schr ̈odinger bridges. While the SOC objectives introduced in Section 3.3 already provide a glimpse of how such optimization can be carried out, the next section will present several concrete algorithms and computational strategies for building stochastic bridges and Schr ̈odinger bridges in practice. 81 4. Building Schr ̈odinger Bridges While the Schr ̈odinger bridge problem provides a variational characterization of the optimal stochastic transport between two marginals, this formulation alone does not immediately reveal how such bridges can be constructed in practice. In this section, we move from the abstract formulation to explore several complementary mechanisms for building stochastic bridges between prescribed endpoint distributions. We begin by interpreting Schr ̈odinger bridges as a mixture of conditional bridges (Section 4.1), which provides an intuitive pathwise construction. We then study time reversal (Section 4.2) and the resulting forward–backward SDE representation (Section 4.3), revealing how drift corrections arise from the score of the evolving distribution. Next, we present Doob’s h-transform (Section 4.4) as a change-of-measure construction that enforces endpoint constraints, followed by an interpretation in terms of Markovian and reciprocal projections (Section 4.5) as entropy-minimizing projections in path space. Finally, we connect these ideas to the stochastic interpolant framework (Section 4.6), which provides practical parameterizations of bridges used in modern generative modeling. Throughout this section, we will use a slight abuse of notation and define the transition density fromX t at timettoX τ at a later timeτ ≥ tunder a path measure asP τ|t (x τ |x t ) or equivalently P τ|t (X τ =x τ |X t =x t ). For instance, the transition density fromxat timettox T at timeTunder the reference measure Q is denoted Q T|t (x T |x) = Q(X T = x T |X t = x). 4.1 Mixture of Conditional Bridges A useful way to understand the structure of Schr ̈odinger bridges is through endpoint conditioning. Rather than viewing the bridge as a single global stochastic process, we can interpret it as a mixture of conditional bridges under the reference processQconnecting specific endpoint pairs (x 0 ,x T ) drawn from a couplingπ 0,T . Each conditional bridge describes the distribution of paths under the reference dynamics conditioned on fixed endpoints, which recovers the optimal Schr ̈odinger bridge when conditioned on the optimal couplingπ ⋆ 0,T that solves the static Schr ̈odinger bridge problem. Proposition 4.1 (Mixture of Endpoint-Conditioned Bridges). Consider the dynamic SB problem with reference process Q and marginal constraints π 0 ,π T ∈P(R d ) defined as: P ⋆ = arg min P u KL (P u ∥Q) : P u 0 = π 0 , P u T = π T (4.1) Then, the unique minimizerP ⋆ can be written as a mixture of endpoint-conditioned bridges Q(·|x 0 ,x T ): P ⋆ (X 0:T ) = Z Q(X 0:T |x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T )(Mixture of Bridges) whereπ ⋆ 0,T is the optimal coupling that solves the static Schr ̈odinger bridge problem with the reference coupling defined as the endpoint law of the reference processQ 0,T given by: π ⋆ 0,T =arg min π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥Q 0,T )(4.2) Proof. By the law of total probability, we can decompose the reference path measure Q as: Q(X 0:T ) = Q(X 0:T |X 0 = x 0 ,X T = x T )Q 0,T (x 0 ,x T )(4.3) By definition, the conditional law of the reference processQ(·|x 0 ,x T ) is the bridge connectingx 0 andx T . Similarly, we can decompose any candidate path measureP u with endpoint lawP 0,T ≡ π 0,T 82 as: P(X 0:T ) = P(X 0:T |X 0 = x 0 ,X T = x T )π 0,T (x 0 ,x T )(4.4) Applying the (KL Divergence Chain Rule), we can split the dynamic SB objective into the KL divergence of the endpoint couplings and conditional KL divergence between the endpoint-conditioned path distributions: KL(P u ∥Q) = KL(π 0,T ∥Q 0,T ) |z (⋆) +E (x 0 ,x T )∼π 0,T KL P(·|x 0 ,x T )∥Q(·|x 0 ,x T ) |z (⋄) (4.5) Clearly, for any fixed endpoint lawπ 0,T the minimizer of (⋄) is achieved whenP(·|x 0 ,x T ) = Q(·|x 0 ,x T ) and the KL is zero. Therefore, solving the dynamic SB over the restricted space of path measures whereP(·|x 0 ,x T ) =Q(·|x 0 ,x T ) reduces to solving the static SB problem which yields the optimal endpoint law: π ⋆ 0,T =arg min π 0,T ∈Π(π 0 ,π T ) KL(π 0,T ∥Q 0,T )(4.6) Since we have shown that the optimalP ⋆ satisfiesP ⋆ (·|x 0 ,x T ) =Q(·|x 0 ,x T ) andP ⋆ 0,T =π ⋆ 0,T , we can conclude that the minimizer is given by: P ⋆ (X 0:T ) = Z Q(X 0:T |x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T ) (4.7) which is unique following proof of uniqueness of the static SB problem in Proposition 1.9. □ We note that this result is a generalization of the result from F ̈ollmer (2006), which states that the solution to the dynamic SB problem with a pure Brownian reference process is given by the mixture of endpoint-conditioned Brownian bridges weighted by the entropic OT plan with quadratic transport cost. Corollary 4.2 (Mixture of Brownian Bridges (F ̈ollmer (2006))). Consider the dynamic SB problem with Brownian reference processQ:dX t =σ t dB t and marginal constraints π 0 ,π T ∈P(R d ). Then, the unique minimizerP ⋆ can be written as a mixture of endpoint- conditioned Brownian bridges Q(·|x 0 ,x T ) given by: P ⋆ (X 0:T ) = Z Q(X 0:T |x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T )(Mixture of Brownian Bridges) whereπ ⋆ 0,T solves the (Entropic OT Problem) with quadratic state costc(x,y) :=∥x−y∥ 2 . This factorized definition of the (Dynamic SB Problem) can be used to define a tractable objective based on conditional stochastic optimal control, which samples pairs from the optimal marginal lawπ ⋆ 0,T and optimizes the interpolating controlled dynamics such that they minimize the KL divergence from the reference drift. Proposition 4.3 (Conditional Stochastic Optimal Control (Proposition 2 in Liu et al. (2023a))). Consider the controlled path measureP u , where the marginal densityp t can be factorized asp t (x) = R p t (x|x 0 ,x T )π 0,T (dx 0 ,dx T ). Given the optimal endpoint distributionπ ⋆ 0,T that solves the static SB problem with reference distributionQ 0,T , the dynamic SB problem objective decomposes into a mixture of conditional stochastic optimal control problems 83 with endpoints sampled from π ⋆ 0,T : inf u E π ⋆ 0,T (dx 0 ,dx T ) Z T 0 E p t (·|x 0 ,x T ) 1 2 ∥u(X t ,t|x 0 ,x T )∥ 2 dt(Conditional SOC Objective) s.t. dX t = (f (X t ,t) + σ t u(X t ,t|x 0 ,x T ))dt + σ t dB t Proof. We recall the dynamic SB problem with the Fokker-Planck constraint as: inf u Z T 0 E p t 1 2 ∥u(X t ,t)∥ 2 dt s.t. ( ∂ t p t =−∇· ((f (x,t) + σ t u(x,t))p t (x)) + σ 2 t 2 ∆p t (x) p 0 = π 0 , p T = π T (4.8) Step 1: Decompose the Minimization Objective.Letπ ⋆ 0,T denote the optimal endpoint coupling solving the static SB problem with reference endpoint lawQ 0,T . Then, the marginal density p t can be factorized as: p ⋆ t (x) = Z p t (x|x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T )(4.9) Under mild regularity assumptions allowing Leibniz’s rule, Fubini’s theorem, and differentiation under the integral sign 13 , we can separate the optimal endpoint lawπ ⋆ 0,T out of the integral, condition on the endpoints, and decompose the objective as: Z T 0 E p t 1 2 ∥u(X t ,t)∥ 2 dt = Z T 0 E π ⋆ 0,T E p t (·|x 0 ,x T ) 1 2 ∥u(X t ,t|x 0 ,x T )∥ 2 dt (law of total expectation) = E π ⋆ 0,T Z T 0 E p t (·|x 0 ,x T ) 1 2 ∥u(X t ,t|x 0 ,x T )∥ 2 dt(Fubini’s theorem) which recovers the conditional SOC objective in (Conditional SOC Objective). Step 2: Derive the Conditional Fokker-Planck Constraint. We still need to show that the conditional dynamicsp t (·|x 0 ,x T ) satisfy the conditional Fokker-Planck constraint. To do this, we substitute the factorization forp ⋆ t in (4.9) into each term of the Fokker-Planck constraint defined as: ∂ t p t (x) | z (⋆) =−∇· ((f (x,t) + σ t u(x,t))p t (x)) |z (⋄) + σ 2 t 2 ∆p t (x) |z (♦) (4.10) Starting off, the time derivative (⋆) can be written as an expectation over the optimal coupling as: ∂ t p t (x) = ∂ t Z p t (x|x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T ) = Z [∂ t p t (x|x 0 ,x T )]π ⋆ 0,T (dx 0 ,dx T ) = E π ⋆ 0,T [∂ t p t (x|x 0 ,x T )](4.11) 13 We assume standard regularity conditions ensuring that expectations and time integrals can be interchanged and that differentiation under the integral sign is valid. Such conditions are typically satisfied for diffusion processes with smooth drift and nondegenerate diffusion coefficients. 84 where the integral can be separated out asπ ⋆ 0,T is not dependent on time. Next, we rearrange the divergence term (⋄) as: ∇· ((f (x,t) + σ t u(x,t))p t (x)) =∇· (f (x,t) + σ t u(x,t)) Z p t (x|x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T ) =∇· Z (f (x,t) + σ t u(x,t))p t (x|x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T ) (linearity of integration) = Z ∇· ((f (x,t) + σ t u(x,t))p t (x|x 0 ,x T ))π ⋆ 0,T (dx 0 ,dx T ) (Leibniz rule) = E π ⋆ 0,T ∇· ((f (x,t) + σ t u(x,t))p t (x|x 0 ,x T )) (4.12) Finally, we write the Laplacian term (♦) as expectation over the optimal coupling given by: ∆p t (x) =∇·∇p t (x) =∇·∇ Z p t (x|x 0 ,x T )π ⋆ 0,T (dx 0 ,dx T ) = Z (∇·∇p t (x|x 0 ,x T ))π ⋆ 0,T (dx 0 ,dx T )(Leibniz rule) = E π ⋆ 0,T [∇·∇p t (x|x 0 ,x T )] = E π ⋆ 0,T [∆p t (x|x 0 ,x T )](4.13) Substituting each of the conditional expressions into (4.10), we get: E π ⋆ 0,T [∂ t p t (x|x 0 ,x T )] |z (⋆) =−E π ⋆ 0,T ∇· ((f (x,t) + σ t u(x,t))p t (x|x 0 ,x T )) | z (⋄) + σ 2 t 2 E π ⋆ 0,T [∆p t (x|x 0 ,x T )] | z (♦) =⇒ E π ⋆ 0,T ∂ t p t (x|x 0 ,x T ) +∇· ((f (x,t) + σ t u(x,t))p t (x|x 0 ,x T ))− σ 2 t 2 ∆p t (x|x 0 ,x T ) = 0 =⇒ ∂ t p t (x|x 0 ,x T ) +∇· ((f (x,t) + σ t u(x,t))p t (x|x 0 ,x T ))− σ 2 t 2 ∆p t (x|x 0 ,x T ) = 0 (Conditional FP Equation) where the last implication follows from the fact that for the expectation to equal zero pointwise for allx, then the terms inside the expectation must equal zero forπ ⋆ 0,T -almost every (x 0 ,x T ). The final line is exactly the (Controlled Fokker-Planck Equation) conditioned on the endpoints (x 0 ,x T ). Since we have shown that the conditional densityp t (·|x 0 ,x T ) satsifies the (Conditional FP Equation), we can conclude that it is a valid density evolution with endpoint constraintsp 0 (·|x 0 ,x T ) =E π ⋆ 0,T [δ x 0 ] andp T (·|x 0 ,x T ) =E π ⋆ 0,T [δ x T ]. By equivalence between the conditional Fokker-Planck equation and the controlled SDE generated byu(X t ,t|x 0 ,x T ), we can also derive the corresponding SDE representation: dX t = (f (X t ,t) + σ t u(X t ,t|x 0 ,x T ))dt + σ t dB t , X 0 = x 0 , X T = x T (4.14) which shows that once the endpoints are sampled from the optimal couplingπ ⋆ 0,T , the (Dynamic SB Problem) decomposes into a mixture of conditional SOC problems.□ This mixture conditional SOC problem decomposes the path space KL minimization to the optimal measure (Mixture of Bridges) from Proposition 4.1 into a static SB problem and a family of conditional bridge problems. This formulation avoids optimizing directly over full path measures and allows us to solve the dynamic SB problem through a tractable two-stage procedure: first, estimate the optimal static coupling (x 0 ,x T )∼ π ⋆ 0,T , then learn the corresponding conditional stochastic control lawu(X t ,t|x 0 ,x T ). We will leverage this idea in Section 6.4 to efficiently learn parameterized Schr ̈odinger bridges with score and flow matching. 85 Figure 6: Time-Reversal of Stochastic Differential Equations. Left: A forward diffusion process overt∈[0,T] evolves from an initial stateX 0 =x 0 under the forward-time SDEdX t , producing marginal densitiesp t (x) that evolve via the Fokker-Planck equation. Right: The corresponding reverse-time stochastic process generates trajectories that traverse the same sequence of marginal densities in the reversed time coordinates=T − t∈[0,T]. Its drift includes an additional score correction∇ logp T−s (x) that accounts for the probability flow of the forward dynamics. This reverse SDE allows trajectories to be simulated starting from the terminal distribution such that they reconstruct the initial distribution. 4.2 Time Reversal A useful approach for constructing stochastic bridges is to analyze the time reversal of a stochastic process, where we leverage the key idea that conditioning on a terminal constraint induces a modified reverse-time dynamics whose drift differs from the original forward drift. Rather than directly enforcing the endpoint condition, we reinterpret the forward process as a backward process and derive the forward Markovian dynamics that reproduce the same marginal density evolution under time reversal, revealing how information about the terminal distribution propagates backward through the SDE dynamics. Consider a forward-time stochastic process defined on the time-horizont∈[0,T] generated from the SDE: dX t = f (X t ,t)dt + σ t dB t , X 0 = x 0 (4.15) One method of building a stochastic bridge that evolves via (4.15) but is conditioned to reach a target stateX T =x T is to reverse the time coordinate of the original forward process such that it can be conditioned on the target state and then, derive a reverse stochastic process that matches density evolution of the forward process with time-reversal. Proposition 4.4 (Time Reversal Formula). Consider a forward-time SDE of the form: dX t = f (X t ,t)dt + σ t dB t (4.16) wheref(x,t) is the drift andσ t is the scalar diffusion coefficient. Letp t (x) denote the marginal density ofX t at timet. Then, the time-reversed process ̃ X s :=X T−s follows the SDE: d ̃ X s = h −f ( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s) i ds + σ T−s d e B s (4.17) 86 where ̃ B s is the Brownian motion with respect to the reverse-time filtration ̃ F=σ( ̃ X τ : 0≤ τ ≤ s). Together, the pair of forward and reverse-time SDEs are given by: ( dX t = f (X t ,t)dt + σ t dB t d ̃ X s = h −f ( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s) i ds + σ T−s d e B s (Time Reversal Formula) Proof. Intuitively, defining an SDE that simulates stochastic processes on the reversed time co- ordinate such that the marginals match those of the forward time stochastic processes requires first reparameterizing the time coordinate of the forward process and then deriving a backward drift that propogates particles on a reversed time coordinate such that it matches the corresponding marginal on the reparameterized forward process. To this end, we break the proof into two steps. Step 1: Define the Forward Process with Time-Reversal. This is a simple reparameterization of the time coordinate such that it followss=T − tthat starts att= 0 ands=Tand terminates att=Tands= 0. We denote this process as ←− X s =X T−t This reparameterization allows us to define a initial condition ←− X 0 =x T and the density ̃p s (x) such thatq(x,0) =π T . With the same drift and diffusion terms, we define the time-reversed process with the SDE: d ←− X s = f ( ←− X s ,T − s) |z := ←− f (X s ,s) ds + σ T−s dB s , ←− X s = x T (4.18) The marginal density p t (x) evolves via the Fokker-Planck equation derived in Section 2.4 as: ∂ t p(x,t) =−∇· (f (x,t)p(x,t)) + σ 2 t 2 ∆p(x,t)(4.19) To write this as an evolution over the time coordinates=T −t, we perform a change of variables as: ∂ ∂s ∂s ∂t |z =−1 p(x,T − s) |z ̃p s (x) =−∇· (f (x,T − s)p(x,T − s) |z ̃p s (x) ) + σ 2 T−s 2 ∆p(x,T − s) |z ̃p s (x) − ∂ ∂s ̃p s (x) =−∇· (f (x,T − s) ̃p s (x)) + σ 2 T−s 2 ∆ ̃p s (x)(4.20) Multiplying both sides by−1, we get that the evolution of time-reversed density ̃p s (x) follows the PDE: ∂ ∂s ̃p s (x) =∇· (f (x,T − s) ̃p s (x))− σ 2 T−s 2 ∆ ̃p s (x)(4.21) Step 2: Derive the Fokker-Planck Equation of the Reverse Stochastic Process. Although reversing the time coordinate defines the density evolution conditioned on a target distribution, it does not tell us the drift and diffusion to follow when simulating paths backward in time since the drift and diffusion are still defined in the forward direction. We define ( ̃ X s ) s∈[0,T ] as the Itˆo process that evolves backward in time. Letb( ̃ X s ,s) be a placeholder for the unknown drift. Since ( ̃ X s ) s∈[0,T ] must be adapted to the backward filtration (F s ) s∈[0,T ] such that it depends only on the past and present given the backward direction, we denote this new F s -adapted Brownian motion asd e B s . To define the diffusion coefficient of the reverse process, we denote ̃σ s ≡ σ T−s . Therefore, we can write the SDE of the backward stochastic process as: d ̃ X s = b( ̃ X s ,s)ds + ̃σ s d e B s (4.22) 87 and the corresponding Fokker-Planck equation as: ∂ ∂s ̃p s (x) =−∇· (b(x,s) ̃p s (x)) + ̃σ 2 s 2 ∆ ̃p s (x)(4.23) Setting (4.23) and (4.21) equal to each other, we have: −∇· (b(x,s) ̃p s (x)) + ̃σ 2 s 2 ∆ ̃p s (x) =∇· (f (x,T − s) ̃p s (x))− ̃σ 2 s 2 ∆ ̃p s (x) −∇· (b(x,s) ̃p s (x))−∇· (f (x,T − s) ̃p s (x)) = ̃σ 2 s ∆ ̃p s (x) ∇· b(x,s) +f (x,T − s) ̃p s (x) = ̃σ 2 s ∆ ̃p s (x)(4.24) By definition, the Laplacian is the divergence of the gradient ∆≡∇·∇, so we can write: ∇· b(x,s) +f (x,T − s) ̃p s (x) =∇· ̃σ 2 s ∇ ̃p s (x) (4.25) Since the terms within the divergences are unknown and we aim to determine an expression for the backward driftb(x,s) with respect to the forward drift and diffusion, we apply a similar technique to the one used to derive the Fokker-Planck equation in Section 2.4 by integrating both sides with an arbitrary test function φ(x). Z R d ∇· b(x,s) +f (x,T − s) ̃p s (x) φ(x)dx = Z R d ∇· ̃σ 2 s ∇ ̃p s (x) φ(x)dx(4.26) Applying the integration by parts identity R R d ∇· f (x)φ(x)dx=− R R d f(x)·∇φ(x)dx 14 to both sides, we have: Z R d b(x,s) +f (x,T − s) ̃p s (x) | z (⋆) ·∇φ(x)dx = Z R d ̃σ 2 s ∇ ̃p s (x) | z (⋄) ·∇φ(x)dx(4.27) Since we definedφ(x) to be any arbitrary test function, (⋆) and (⋄) must be equal to each other for all x, and thus we can write: b(x,s) +f (x,T − s) ̃p s (x) = ̃σ 2 s ∇ ̃p s (x)(4.28) Rearranging and dividing both sides by ̃p s (x)≥ 0, we have: b(x,s) ̃p s (x) =−f (x,T − s) ̃p s (x) + ̃σ 2 s ∇ ̃p s (x) b(x,s) =−f (x,T − s) + ̃σ 2 s ∇ ̃p s (x) ̃p s (x) |z score function (4.29) Since ∇ ̃p s (x) ̃p s (x) =∇ log ̃p s (x) which is the score function of the density ̃p s (x), we can write the final backward drift as: b(x,s) =−f (x,T − s) + ̃σ 2 s ∇ log ̃p s (x)(4.30) 14 which assumesfis continuously differentiable andφis smooth with sufficient decay at infinity such that the boundary term from integration by parts vanishes. 88 which can also be expressed in terms of the forward density ̃p s (x) =p(x,T−s) and forward diffusion ̃σ s = σ T−s as: b(x,s) =−f (x,T − s) + σ 2 T−s ∇ logp(x,T − s)(4.31) Substitutingb(x,s) into (4.23), we get the Fokker-Planck equation satisfied by the backward stochastic process (or bridge) as: ∂ ∂s p T−s (x) =−∇· (−f (x,T − s) + σ 2 T−s ∇ logp T−s (x))p T−s (x) + σ 2 T−s 2 ∆p T−s (x)(4.32) and the corresponding SDE given by: d ̃ X s = h −f ( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s) i ds + σ T−s d e B s which completes the pair of forward and reverse-time SDEs: ( dX t = f (X t ,t)dt + σ t dB t d ̃ X s = h −f ( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s) i ds + σ T−s d e B s (4.33) and we conclude our proof.□ Using (Time Reversal Formula), denoising diffusion can be viewed as a special case of the Schr ̈odinger bridge problem where the reference process is a variance-explooding SDE with zero drift. Denoising Diffusion as a Special Case of the Schr ̈odinger Bridge Here, we will show that Schr ̈odinger bridges are a generalization of score-based diffusion models introduced in (Huang et al., 2021; Song et al., 2019; 2020a; b). Specifically, score-based models are a special case where the forward-time stochastic process is the variance exploding SDE with zero-drift: dX t = σ t dB t , X 0 = x 0 ∼ p data (4.34) where the variance of the noise accumulates over time followingβ t := R t 0 σ 2 s ds . Applying the time reversal described above, we can derive the SDE of the backward stochastic process ̃ X s = X T−s as: d ̃ X s = −f (x,T − s) |z =0 (no drift) +σ 2 T−s ∇ log ̃p s (x) ds + σ T−s d e B s = σ 2 T−s ∇ log ̃p s (x) ds + σ T−s d e B s (4.35) where the initial distribution is a Gaussian with large variance ̃ X 0 ∼N(x 0 ,β T ). Since the variance of the forward process isβ t := R t 0 σ 2 s ds, we can derive the variance of the backward process as the total variance subtracted by the variance reduced in the backward process up to timesgiven by ̃ β s =β T − β s . Therefore, the density of the backward process can be written explicitly as a Gaussian with meanxand variance ̃ β s , denoted ̃p s (x) =N(x 0 ,β T − β s ). Then, the score function can be 89 rewritten as: ̃p s (x) = 1 (2π(β T − β s )) d/2 exp − ∥x−x 0 ∥ 2 2(β T − β s ) log ̃p s (x) =− d 2 log(2π(β T − β s ))− 1 2(β T − β s ) ∥x−x 0 ∥ 2 ∇ log ̃p s (x) =− 1 2(β T − β s ) ∇ ∥x−x 0 ∥ 2 =− 1 2(β T − β s ) · 2(x−x 0 ) =− x−x 0 β T − β s (4.36) Substituting this expression for the score into the backward SDE, we get the SDE of a Brownian bridge: d ̃ X s = σ 2 T−s x 0 − ̃ X s β T − β s ds + σ T−s d e B s (4.37) In practice, we do not know clean samplesx 0 ∼ p data during the backward process, so instead of initializing the distribution from ̃ X 0 ∼ N(x 0 ,β T I d ), we approximate it with ̃ X 0 ∼N (0,β T I d ), which works when β T is large relative to the data variance. We can also define the time-reversal formula for a controlled SDE by simply replacing the drift f (X t ,t) with a controlled drift (f (X t ,t) + σ t u(X t ,t)) and following the same derivation. Corollary 4.5 (Time Reversal for Controlled SDEs). Consider a forward-time controlled SDE of the form: dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB t (4.38) whereu(x,t) is the control drift andσ t is the scalar diffusion coefficient. Letp t (x) denote the marginal density ofX t at timet. Then, the time-reversed process ̃ X s :=X T−s follows the SDE: d ̃ X s = h −f ( ̃ X s ,T − s)− σ T−s u( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s) i ds + σ T−s d e B s (4.39) and the pair of forward and reverse-time SDEs is given by: ( dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB t d ̃ X s = h −f ( ̃ X s ,T − s)− σ T−s u( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s) i ds + σ T−s d e B s (4.40) The key takeaway from this section is that reversing a forward-time stochastic process is not as simple as inverting the time coordinate of the trajectories. While the forward process propagates densitiesp t according to its driftf(X t ,t) and diffusion coefficients, running the process backward requires deriving a new stochastic process whose drift correctly propagates the marginal densities in the reverse time direction such that it reconstructs the same marginal densities as the forward process s = T − t∈ [0,T ]. The final (Time Reversal Formula) reveals that the reverse-time drift includes the negative forward drift −f(X s ,T−s) and an additional correction proportional to the score function∇ logp T−s (x), which compensates for the spreading of probability density from the forward diffusion. This insight is fundamental to modern generative modeling frameworks, such as score-based diffusion models and 90 Figure 7: Time Reversal vs. Forward-Backward SDE. Top: Reversing an uncontrolled forward diffusion requires correcting the drift using the score function∇ logp t (x), which depends on the evolving marginal density and can lead to unstable or uncontrolled trajectories. Bottom: In the more general Schr ̈odinger bridge setting, the forward process evolves via a control driftu(x,t) :=σ t ∇ x logφ t (x) which contains the forward Schr ̈odinger potentialφ t . This controlled forward process induces a corresponding backward SDE with the controlσ t ∇ x log ˆφ t (x), containing the backward Schr ̈odinger potentialˆφ t . This enables accurate transport between two structured distributions with optimal bridge dynamics. Schr ̈odinger bridge methods, where learning the score function enables simulation of reverse-time dynamics and thus the generation of samples from complex target distributions. 4.3 Forward-Backward Stochastic Differential Equations A key limitation of the standard time-reversal formula from Section 4.2 is that it assumes an uncontrolled forward SDE, where the driftf(x,t) is deterministic and can be easily simulated in either direction, such as denoising diffusion, where the forward process is a known variance-exploding SDE. In the general Schr ̈odinger bridge setting, however, the goal is to interpolate between two structured distributions, and the optimal forward drift that transportsπ 0 toπ T is itself unknown and must be solved for as part of the optimization problem. As a result, (Time Reversal Formula) is no longer sufficient on its own, and we must instead derive a coupled pair of forward-backward SDEs that characterizes both the forward and backward controlled dynamics. Recall the (Dynamic SB Problem), where the optimal control and optimal marginal density (u ⋆ ,p ⋆ t ) are defined by the non-linear (HJB-FP System) as: ( ∂ t ψ t + σ 2 t 2 ∥∇ψ t ∥ 2 +⟨∇ψ t ,f⟩ =−σ 2 t ∆ψ t ∂ t p ⋆ t +∇· (p ⋆ t (f + σ 2 t ∇ψ)) = σ 2 t ∆p ⋆ t s.t. ( p ⋆ 0 (x) = π 0 p ⋆ T (x) = π T (4.41) While the Hopf-Cole transform described in Section 2.8 transforms the non-linear PDEs into linear PDEs, solving them remains challenging. In Section 6.1, we show that score-based generative modeling can be derived from the time reversal of a forward SDE described in Section 4.2. This naturally leads to the question: how can we turn the PDEs defining the solution to the dynamic SB problem into SDEs that can be solved with likelihood training? To answer this, we can leverage the theory of forward-backward SDEs (Exarchos et al., 2018). This theory extends the Feynman-Kac theory from Section 2.4, which represents solutions of certain linear 91 parabolic PDEs as expectations over trajectories of a forward stochastic process. Forward–backward SDE theory generalizes this idea by introducing a coupled system consisting of a forward SDE that generates trajectories and a backward stochastic process that evolves along those trajectories and encodes the solution of the PDE. Forward-Backward SDE Theory Consider a functionψ t (x) :R d ×[0,T]→ R∈ C 2,1 (R d ×[0,T]) and a parabolic PDE of the form: ∂ t ψ t + σ 2 t 2 ∆ψ t +⟨f,∇ψ t ⟩ +h(x,t,ψ t ,σ t ∇ψ t ) = 0, ψ T (x) = Φ(x)(4.42) whereψ T (x) = Φ(x) is the terminal constraint. Suppose the stateX t =xis a stochastic process X 0:T that evolves via a forward SDE defined as: dX t = f (X t ,t)dt + σ t dB t , X 0 = x 0 (4.43) Then, we can define the stochastic processY t :=ψ t (X t ) andZ t :=σ t ∇ψ t (X t ) which evolves via the backward SDE defined as: dY t =−h(X t ,t,Y t ,Z t )dt +Z ⊤ t dB t , Y T = Φ(X T )(4.44) such that the solutionψ t to (4.42) is equivalent to the solution to the forward-backward SDEs defined in (4.43) and (4.44). While this gives us a way to convert a single non-linear PDE into a pair of FBSDEs, the dynamic SB problem with arbitrary reference dynamics is defined by a pair of coupled PDEs given in Section 2.8 by the Hopf-Cole transform. Therefore, we must derive a system of forward-backward SDEs that describes the evolution of both potentials. Forward-Backward SDEs for Dynamic Schr ̈odinger Bridge In Section 2.8, we transformed the non-linear (HJB-FP System) with the Hopf-Cole trans- form to get the system of linear PDEs that defines the solution to the Schr ̈odinger bridge problem with arbitrary reference dynamics: ( ∂ t φ t +⟨∇φ t ,f⟩ =− σ 2 t 2 ∆φ t ∂ t ˆφ t +∇· ( ˆφ t f ) = σ 2 t 2 ∆ ˆφ t s.t. ( p ⋆ 0 = φ 0 ˆφ 0 p ⋆ T = φ T ˆφ T (4.45) whre (φ t , ˆφ t ) are the pair of Schr ̈odinger potentials that uniquely characterize the optimal control and optimal state PDF (u ⋆ ,p ⋆ t ) given by: u ⋆ (x,t) = σ t ∇ logφ t (x)(4.46) p ⋆ t (x) = φ t (x) ˆφ t (x)(4.47) Given the stochastic processX 0:T , we can define the random variables associated with the forward and backward potentials as: Y t ≡ Y t (X t ,t) = logφ t (X t ),Z t ≡ Z t (X t ,t) = σ t ∇ logφ t (X t )(4.48) b Y t ≡ b Y t (X t ,t) = log ˆφ t (X t ), b Z t ≡ b Z t (X t ,t) = σ t ∇ log ˆφ t (X t )(4.49) 92 Then, the forward-backward SDEs that define the evolution of these random variables are given by: dX t = (f (X t ,t) + σ t Z t )dt + σ t dB t dY t = 1 2 ∥Z t ∥ 2 dt +Z ⊤ t dB t d b Y t = ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 +Z ⊤ t b Z t dt + b Z ⊤ t dB t s.t. X 0 = x 0 Y T + b Y T = logπ T (X T ) Y t + b Y t = logp ⋆ t (X t ) u ⋆ (X t ,t) = Z t (4.50) where the solution to the FBSDEs in (4.50) is the solution to the linear PDEs in (4.45). Derivation. Step 1: Derive the Stochastic Processes that Encode Optimality Conditions. Recall from Theorem 2.24 that the dynamic SB problem has the control driftu ⋆ (x,t) =σ t ∇ logφ t (x) which defines the forward SDE: dX t = f (X t ,t) + σ 2 t ∇ logφ t (X t ) dt + σ t dB t (4.51) Furthermore, the optimal marginal densityp ⋆ t (x) is defined by both the forward and backward potentials p ⋆ t (x) = φ t (x) ˆφ t (x), which can be transformed to a logarithm as: logp ⋆ t (x) = logφ t (x) + log ˆφ t (x)(4.52) Therefore, both the optimal control drift and marginal density can be expressed as the logarithm of the SB potentials (φ t , ˆφ t ). The evolution oflogφ t andlog ˆφ t along the stochastic trajectoriesX 0:T can be described by a pair of stochastic processes (Y t ) t∈[0,T ] and ( b Y t ) t∈[0,T ] defined as: ( Y t := logφ t (X t ) b Y t := log ˆφ t (X t ) s.t.logp ⋆ t (X t ) = Y t + b Y t (4.53) Unlike score-based generative modeling, the score function∇ logq t (x) is replaced with the gradient of the log potentials∇ logφ t (x) and∇ log ˆφ t (x). Since the gradient of a random variable defined by a stochastic process is also a random variable, we define the auxillary stochastic processes (Z t ) t∈[0,T ] and ( b Z t ) t∈[0,T ] which track the gradient of the log potentials: ( Z t := σ t ∇ logφ t (X t ) b Z t := σ t ∇ log ˆφ t (X t ) s.t. u ⋆ (X t ,t) = Z t (4.54) which transforms the forward SDE into: dX t = [f (X t ,t) + σ t Z t ]dt + σ t dB t (4.55) Now that we have defined the time-indexed random variables (Y t , b Y t ) and (Z t , b Z t ) that encode the optimality constraints, we can derive the corresponding forward-backward SDEs. Step 2: Derive the Forward-Backward SDEs. To determine the time-evolution ofY t and b Y t , we apply (Itˆo’s Formula). For simplicity, we will drop explicit (X t ,t) dependence and denote φ t ≡ φ t (X t ) andf ≡ f(X t ,t) throughout the derivation. Starting withY t =logφ t (X t ), we apply Itˆo’s formula to get the SDE: d logφ t = ∂ t logφ t +⟨f + σ 2 t ∇ logφ t ,∇ logφ t ⟩ + σ 2 t 2 ∆ logφ t dt +∇ logφ ⊤ t σ t dB t = ∂ t logφ t |z given by PDE +⟨f,∇ logφ t ⟩ + σ 2 t ∥∇ logφ t ∥ 2 + σ 2 t 2 ∆ logφ t dt +∇ logφ ⊤ t σ t dB t (4.56) 93 From the (Hopf-Cole PDEs), we have that the potential functionφ t evolves via the linear PDE given by: ∂ t φ t =−⟨∇φ t ,f⟩− σ 2 t 2 ∆φ t (4.57) Using the chain rule, we have∂ t logφ t = ∂ logφ t ∂φ t ∂ t φ t , so the PDE gives us the following form of the time evolution for logφ t : ∂ t logφ t = ∂ logφ t ∂φ t ∂ t φ t = 1 φ t −⟨∇φ t ,f⟩− σ 2 t 2 ∆φ t =− ∇φ t φ |z =∇ logφ t ,f − σ 2 t 2 1 φ t ∆φ t =−⟨∇ logφ t ,f⟩− σ 2 t 2 ∆ logφ t (4.58) Substituting the expression for ∂ t logφ t into the SDE (4.56), we get: d logφ t = ∂ t logφ t | z given by PDE +⟨f,∇ logφ t ⟩ + σ 2 t ∥∇ logφ t ∥ 2 + σ 2 t 2 ∆ logφ t dt + (σ t ∇ logφ t ) ⊤ dB t = −⟨∇ logφ t ,f⟩− σ 2 t 2 ∆ logφ t +⟨f,∇ logφ t ⟩ + σ 2 t ∥∇ logφ t ∥ 2 + σ 2 t 2 ∆ logφ t dt + (σ t ∇ logφ t ) ⊤ dB t =∥σ 2 t ∇ logφ t | z =:Z t ∥ 2 dt + (σ t ∇ logφ t ) ⊤ |z =:Z ⊤ t dB t = 1 2 ∥Z t ∥ 2 dt +Z ⊤ t dB t (4.59) which is exactly the SDE for Y t : dY t = 1 2 ∥Z t ∥ 2 dt +Z ⊤ t dB t (4.60) Now, we apply Itˆo’s formula to b Y t = log ˆφ t (X t ) to get: d log ˆφ t (X t ,t) = ∂ t log ˆφ t + (f + σ 2 t ∇ logφ t ) ⊤ ∇ log ˆφ t + σ 2 t 2 ∆ log ˆφ t dt + (σ t ∇ log ˆφ t ) ⊤ dB t = ∂ t log ˆφ t | z given by PDE +⟨f,∇ log ˆφ t ⟩ +⟨σ 2 t ∇ logφ t ,∇ log ˆφ t ⟩ + σ 2 t 2 ∆ log ˆφ t dt + (σ t ∇ log ˆφ t ) ⊤ dB t (4.61) Similarly toφ t , the (Hopf-Cole PDEs) defines the time evolution ofˆφvia the following linear PDE: ∂ t ˆφ t =−∇· ( ˆφ t f ) + σ 2 t 2 ∆ ˆφ t (4.62) Applying the chain rule, we have: ∂ t log ˆφ t = ∂ log ˆφ t ∂ ˆφ t ∂ t ˆφ t = 1 ˆφ t −∇· ( ˆφ t f ) + σ 2 t 2 ∆ ˆφ t =− 1 ˆφ t (∇ ˆφ t | z ∇ log ˆφ t ·f )− 1 ˆφ t ˆφ t ∇·f + σ 2 2 ∆ ˆφ t ˆφ t =−⟨∇ log ˆφ t ,f⟩−∇·f + σ 2 t 2 ∆ ˆφ t ˆφ t (4.63) 94 Substituting this expression into the SDE (4.61), we can cancel like terms to get: d log ˆφ t (X t ) = −⟨∇ log ˆφ t ,f⟩−∇·f + σ 2 t 2 ∆ ˆφ t ˆφ t +⟨f,∇ log ˆφ t ⟩ +⟨σ 2 t ∇ logφ t ,∇ log ˆφ t ⟩ + σ 2 t 2 ∆ log ˆφ t dt + (σ t ∇ log ˆφ t ) ⊤ dB t = −∇·f + σ 2 t 2 ∆ ˆφ t ˆφ t +⟨σ 2 t ∇ logφ t ,∇ log ˆφ t ⟩ + σ 2 t 2 ∆ log ˆφ t dt + (σ t ∇ log ˆφ t ) ⊤ dB t (4.64) We can expand the Laplacian using the product rule: ∆ log ˆφ t =∇· ( ˆφ −1 t ∇ ˆφ t ) =∇( ˆφ −1 t ) ⊤ ∇ ˆφ t + ˆφ −1 t ∆ ˆφ t = (− ˆφ −2 t ∇ ˆφ t ) ⊤ ∇ ˆφ t + ˆφ −1 t ∆ ˆφ t =− ∥∇ ˆφ t ∥ 2 φ 2 t + ∆ ˆφ t ˆφ t =−∥∇ log ˆφ t ∥ 2 + ∆ ˆφ t ˆφ t (4.65) =⇒ ∆ ˆφ t ˆφ t = ∆ log ˆφ t +∥∇ log ˆφ t ∥ 2 (4.66) Substituting the expression for ∆ ˆφ t ˆφ t back into (4.64), we get: d log ˆφ t (X t ) = −∇·f + σ 2 t 2 ∆ log ˆφ t +∥∇ log ˆφ t ∥ 2 +⟨σ 2 t ∇ logφ t ,∇ log ˆφ t ⟩ + σ 2 t 2 ∆ log ˆφ t dt + (σ t ∇ log ˆφ t ) ⊤ dB t = −∇·f + σ 2 t ∆ log ˆφ t | z ∇·(σ 2 t ∇ log ˆφ t −f ) + σ 2 t 2 ∥∇ log ˆφ t ∥ 2 +⟨σ 2 t ∇ logφ t ,∇ log ˆφ t ⟩ dt + (σ t ∇ log ˆφ t ) ⊤ dB t = ∇· (σ 2 t ∇ log ˆφ t | z =:σ t b Z t −f ) + 1 2 ∥σ t ∇ log ˆφ t |z =: b Z t ∥ 2 +⟨σ t ∇ logφ t |z =:Z t ,σ t ∇ log ˆφ t |z =: b Z t ⟩ dt + (σ t ∇ log ˆφ t ) ⊤ |z =: b Z ⊤ t dB t = ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 +Z ⊤ t b Z t dt + b Z ⊤ t dB t (4.67) which gives us the SDE for b Y t : d b Y t = ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 +Z ⊤ t b Z t dt + b Z ⊤ t dB t (4.68) Finally, by combining the three SDEs, we have shown that the solution to the dynamic SB problem is equivalent to the solution to the system of three FBSDEs. dX t = (f (X t ,t) + σ t Z t )dt + σ t dB t d ̃ X s = (−f ( ̃ X s ,T − s) + σ T−s b Z s )ds + σ T−s d ̃ B s dY t = 1 2 ∥Z t ∥ 2 dt +Z ⊤ t dB t d b Y t = ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 +Z ⊤ t b Z t dt + b Z ⊤ t dB t (Forward-Backward SDEs) which describe the evolution of the Schr ̈odinger bridge potentials (φ t , ˆφ t ) that characterize the optimal control driftu ⋆ (X t ,t) =Z t and marginal densitylogp ⋆ t (X t ) =Y t + b Y t as SDEs that can be simulated forward and backward in time.□ We now connect the FBSDE representation to the time-reversal representation and show that the Schr ̈odinger bridge potentials yield a generalization of the time-reversal strategy to settings where the forward SDE has an additional non-deterministic control drift σ t u(x,t). 95 Remark 4.6 (Forward-Backward SDE Theory Generalizes the Time-Reversal Formula). The forward–backward SDE formulation provides a natural generalization of the classical time-reversal formula for uncontrolled forward SDEs. In the special case where the forward dynamics contain no control drift (i.e.,Z t ≡0), the forward SDE reduces to the reference diffusion, and the backward control b Z t becomes proportional to the score of the marginal density p ⋆ t : Y t = logφ t (X t )≡ 0 =⇒ logp ⋆ t (X t ) = 0 + b Y t = log ˆφ t (X t ) =⇒ b Z t = σ t ∇ log ˆφ t (X t ) = σ t ∇ logp ⋆ t (X t )(4.69) which yields a reverse-time SDE that is exactly the time-reversal SDE derived in (Time Reversal Formula): d ̃ X s = h −f ( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s) i ds + σ T−s d e B s (4.70) that characterize the generation process of score-based diffusion models. Therefore, we can interpret forward-backward SDEs as a generalization of the time-reversal formula and the dynamic SB problem as a generalization of score-based diffusion. In Section 6.2, we will turn this FBSDE construction into a tractable likelihood-based training objective. However, before introducing any training formulations, we continue our exploration of building Schr ̈odinger bridges. Next, we discuss endpoint conditioning via Doob’sh-transform, which provides additional insight into the structure of Schr ̈odinger bridges. 4.4 Doob’s h-Transform The central goal when solving the Schr ̈odinger bridge problem is to derive a controlled stochastic process by minimally shaping a reference stochastic process that originates an initial distribution π 0 by conditioning it on a terminal distributionπ T . One way to achieve this conditioned process is using Doob’sh-Transform (Rogers et al., 2000; S ̈arkk ̈a et al., 2019), which introduces a tilting function, known as theh-function, such that multiplying the transition density of the reference stochastic process by theh-function and re-normalizing produces the transition density of the optimal Schr ̈odinger bridge. Proposition 4.7 (Doob’sh-Transform). Under a reference stochastic processQ, letQ(X τ = y|X t =x) denote the transition kernel fromxatttoyat timeτ ≥ t. Define a function h(x,t) : R d × [0,T ]→ R that satisfies the following space-time Markov consistency property: h(x,t) = E Q [h(X τ ,τ )|X t = x] = Z R d Q(X τ = y|X t = x)h(y,τ )dy(h-Function) Then, we define the stochastic process P h by tilting the reference process Q as: P h (X τ = y|X t = x) = Q(X τ = y|X t = x) h(y,τ ) h(x,t) (Doob’s h-Transform) whereP h (X τ =y|X t =x) is the tilted transition kernel ofP h . Then,P h is Markov and has the associated SDE: dX t = f (x,t) + σ 2 t ∇ logh(x,t) dt + σ t dB t (Doob’s h-Transform SDE) 96 Proof. First, we show that the tilted transition density integrates to one: Z R d P h (X τ = y|X t = x)dy = Z R d Q(X τ = y|X t = x) h(y,τ ) h(x,t) dy = 1 h(x,t) Z R d Q(X τ = y|X t = x)h(y,τ )dy | z =:h(x,t) = 1(4.71) which proves thatP h is a valid Markov process which depends only onX t . Next, we will derive the corresponding SDE that generatesP h by defining a smooth test functionφ(x,t) :R d ×[0,T]→ R∈ C 2,1 (R d ×[0,T]) and derive the expression for the generatorA t and using Itˆo’s formula to recover the expression for the drift fieldv(x,t). By definition, the generator of a stochastic process is given by: (A h t φ)(x) := lim ∆t→0 E h [φ(X t+∆t ,t + ∆t)|X t = x]− φ(x) ∆t (4.72) From (Doob’sh-Transform), we have that the expectation of the test function evaluated on the tilted measure can be written as: E h [φ(X t+∆t ,t + ∆t)|X t = x] := 1 h(x,t) E[φ(X t+∆t ,t + ∆t)h(X t+∆t ,t + ∆t)|X t = x](4.73) which can be substituted back into (4.72) to get: (A h t φ)(X t ) = lim ∆t→0 E[φ(X t+∆t ,t + ∆t)h(X t+∆t ,t + ∆t)|X t = x]− φ(x)h(x,t) h(x,t)∆t = 1 h(x,t) lim ∆t→0 E[φ(X t+∆t ,t + ∆t)h(X t+∆t ,t + ∆t)|X t = x]− φ(x)h(x,t) ∆t | z =:A t (φ(x)h(x,t)) = 1 h(x,t) A t (φ(x)h(x,t))(4.74) whereA t is the generator of the untilted reference process defined in (Uncontrolled Generator). Now, using the expanded form of the generator derived in (Uncontrolled Generator) and denoting φ≡ φ(x,t) and h≡ h(x,t) for simplicity, we have: A h t φ = 1 h ∂ t (φh) +⟨f,∇(φh)⟩ + σ 2 t 2 ∆(φh) = 1 h φ∂ t h +⟨f,h∇φ⟩ +⟨f,φ∇h⟩ + σ 2 t 2 ∇· (h∇φ + φ∇h) = 1 h φ∂ t h + h⟨f,∇φ⟩ + φ⟨f,∇h⟩ + σ 2 t 2 (h∆φ + φ∆h + 2∇φ·∇h) (4.75) 97 Since, we defined theh-function as a conditional expectationh(x,t) =E Q [h(X τ ,τ)|X t =x], it satisfies the (Martingale Property), and the expression for the unconditional generator vanishes: A h t φ = 1 h φ ∂ t h +⟨f,∇h⟩ + σ 2 t 2 ∆h |z =A t h=0 (h is a Martingale) + 1 h h⟨f,∇φ⟩ + σ 2 t 2 (h∆φ + 2∇φ·∇h) =⟨f,∇φ⟩ + σ 2 t 2 ∆φ + σ 2 t ∇φ· ∇h h |z ∇ logh =⟨f,∇φ⟩ + σ 2 t 2 ∆φ +⟨∇φ,σ 2 t ∇ logh⟩ =⟨f + σ 2 t ∇ logh |z v(x,t) ,∇φ⟩ + σ 2 t 2 ∆φ(4.76) which means the control drift of the Doobh-transformed process is defined asv(x,t) :=f(x,t) + σ 2 t ∇ logh(x,t) and the corresponding SDE is: dX t = f (x,t) + σ 2 t ∇ logh(x,t) dt + σ t dB t (Doob’s h-Transform SDE) and we conclude our proof.□ The previous derivation explicitly shows how reweighting a reference diffusion with a positive space–time functionh(x,t) modifies the infinitesimal generator of the process. The following Corollary summarizes this result and provides an expression for the generator of the reweighted path measure in compact form. Corollary 4.8 (Generator of Reweighted Path Measure). LetQbe a reference path measure with infinitesimal generatorA t andh(x,t) :R d ×[0,T]→ Rbe theh-function defined in (h-Function). Then, the path measure reweighted byhis theh-transform ofQwith generator defined as: A h t φ(x) := A t (φ(x)h(x,t))− φ(x)A t h(x,t) h(x,t) =A t φ(x) +⟨σ 2 t ∇ logh(x,t),∇φ(x)⟩(4.77) Therefore, Doob’sh-transform provides a precise mechanism for incorporating endpoint information into the dynamics of a reference process. This mechanism allows us to define anh-function that exactly recovers the Markov dynamics of the optimal Schr ̈odinger bridge. Corollary 4.9 (Schr ̈odinger Bridge as Doob’sh-Transform). Given the Schr ̈odinger potentials (φ t , ˆφ t ) that generate the solution to the (Dynamic SB Problem), we can define theh-function as: h(x,t) := φ t (x) = E Q [φ T (X T )|X t = x] = Z R d Q(X T = y|X t = x)φ T (X T )dy (h-Function for SB) Then, the optimal Schr ̈odinger bridge path measureP ⋆ is the Doob’sh-transform ofQ, where for any 0≤ t≤ τ ≤ T , we have: P ⋆ (X τ = y|X t = x) = Q(X τ = y|X t = x) h(y,τ ) h(x,t) (Doob’s h-Transform for SB) Equivalently, P ⋆ is Markov and has the associated SDE: dX t = f (x,t) + σ 2 t ∇ logh(x,t) dt + σ t dB t (Doob’s h-Transform SDE) 98 Proof Sketch. From the derivation of the (Schr ̈odinger Potentials), the optimal path measure satisfies: dP ⋆ dQ (X 0:T ) = ˆφ 0 (X 0 )φ T (X T )(4.78) Conditioning onX t =xand using the Markov property ofQ, we observe that the future evolution from time t to τ depends only on the terminal reweighting through: φ t (x) = E Q [φ T (X T )|X t = x](4.79) Therefore, the conditional transition kernel underP ⋆ is obtained by tilting the reference kernel by the ratio φ τ (y) φ t (x) , which is exactly Doob’s h-transform.□ This perspective shows that the Schr ̈odinger bridge corresponds to a precise reweighting the reference processQthrough a harmonic functionh(x,t) :=φ t (x) which modifies the forward drift byσ 2 t ∇ logh(x,t). The resulting process therefore evolves according to the controlled drift f(x,t) +σ 2 t ∇ logh(x,t), which can be interpreted as the minimal modification of the reference dynamics required to enforce the desired endpoint constraints. 4.5 Markovian and Reciprocal Projections In many settings, the Schr ̈odinger bridge yields a path measure whose dependencies span the entire trajectory, making direct simulation difficult. This raises a natural question: Can we approximate a general bridge by a Markov process that is easier to simulate, while remaining as close as possible in relative entropy? The Markovian projection provides a principled answer to this question. Given an arbitrary path measure, we project it onto the space of Markov measuresMby minimizing KL divergence over processes whose future states depend only on the present state. This produces the closest Markov approximation to the original bridge. Formally, we define the space of Markov measures as: M := M∈P(C([0,T ]; R d ))|∀0≤ s < t≤ T, E M [f (X t )|X s ] = E M [f (X t )|F s ] (4.80) where F s denotes the filtration generated by the process up to time s (Definition 2.3). In the Schr ̈odinger bridge setting, we seek to construct a stochastic bridge between empirical endpoint distributionsπ 0 andπ T that is close in relative entropy to a reference measureQdefined by the SDE: Q : dX t = f (X t ,t)dt + σ t dB t (Reference SDE) Given a couplingπ 0,T ∈P(R d × R d ) between the endpoint distributions, we will consider the path measure generated by mixture of endpoint-conditioned bridges defined as: Π = π 0,T Q ·|0,T ∈P(C([0,T ]; R d ))(Mixture of Bridges) which can be interpreted as constructing a bridge by first sampling endpoints (x 0 ,x T )∼Π 0,T and then generating the intermediate bridge dynamics according to the reference path measure conditioned on the endpoints. For fixed (x 0 ,x T ), the corresponding conditional bridge dynamics under the reference measure Q are given by the (Doob’s h-Transform SDE) defined as: Q ·|0,T (·|x 0 ,x T ) : dX t = f (X t ,t) + σ 2 t ∇ log Q T|t (x T |X t ) dt + σ t dB t (Bridge SDE) whereQ T|t (x T |X t ) is the probability of reachingx T from the current stateX t via the SDE ofQ. Since the drift in (Bridge SDE) is conditioned on the future endpointx T , it is not Markov with respect toX t . Therefore, simulating the SDE requires evaluating the transition densityQ T|t (x T |X t ) at every step, which in general is computationally intractable. 99 To obtain a tractable process, we instead construct the Markovian projection of the bridge measure Π, which is the Markov process whose drift depends only on the current stateX t and that minimizes the KL divergence to the original bridge measure. Proposition 4.10 (Markovian Projection (Proposition 2 in Shi et al. (2023))). Consider a mixture of bridges Π = Π 0,T Q ·|0,T that bridges distributionsπ 0 andπ T via the endpoint law Π 0,T , where each conditional bridge is defined by (Bridge SDE) generated from the reference measure. Then, the Markovian projection of Π is denoted: M ⋆ := proj M (Π)∈M(Markovian Projection) with the associated SDE: M ⋆ : dX t = [f (X t ,t) + σ t u ⋆ (X t ,t)]dt + σ t dB t (Markovian Projection SDE) s.t. u ⋆ (x,t) = σ t E Π T|t ∇ log Q T|t (X T |X t )|X t = x (Markovian Projection Drift) where σ t > 0. Then, M ⋆ satisfies the following properties: (i) It is the Markov measureM ⋆ ∈Mthat minimizes the reverse KL divergence with the mixture of bridges Π such that: M ⋆ = arg min M KL (Π|M) : M∈M(4.81) KL (Π|M) = 1 2 Z T 0 E Π 0,t σ t E Π T|0,t ∇ log Q T|t (X T |X t )|X 0 ,X T −u ⋆ (X t ,t) 2 (4.82) (i) It preserves the time marginals of Π t for all t∈ [0,T ] such that: ∀t∈ [0,T ], M ⋆ t = Π t (4.83) Proof. We break down the proof in three steps. Step 1: Derive the Optimal Control Drift of the Markovian Projection. Recall that the bridge path measure can be obtained by reweighting the reference measureQby the Radon–Nikodym derivative of the endpoint coupling Π 0,T with respect to the reference endpoint law Q 0,T : π 0,T (x 0 ,x T ) := dΠ 0,T dQ 0,T (x 0 ,x T ) =⇒ Π(X 0:T ) = π 0,T (X 0 ,X T )Q(X 0:T )(4.84) Conditioning on the initial state X 0 = x 0 , the corresponding conditional bridge law becomes: Π(X 0:T |X 0 = x 0 ) = h(X T ,T )Q(X 0:T |X 0 = x 0 ),where h(X T ,T ) := dΠ(X T |X 0 = x 0 ) dQ(X T |X 0 = x 0 ) (4.85) This means that given an initial statex 0 , the bridge law is obtained by reweighting the reference diffusion by a terminal weight. This is exactly the (h-Function) from (Doob’sh-Transform) for time t→ T with h-function defined as: h(x,t) = Z R d Q T|t (X T = x T |X t = x)h(X T ,T )dx T = E Q [h(X T ,T )|X t = x,X 0 = x 0 ] (Doob’s h-transform) which is the conditional expectation of the endpoint weight under the reference process. Corollary 4.8 establishes the fact that reweighting paths fromQby theh-function changes the generator by adding the drift⟨σ 2 t ∇ logh(x,t),∇f(x,t)⟩which is still Markov. Therefore, the bridge measure Π ·|0=x 0 conditioned on X 0 = x 0 follows the SDE: dX t = f (X t ,t) + σ 2 t ∇ x logh(X t ,t) dt + σ t dB t (4.86) 100 To derive the form of the correction term∇ logh(X t ,t) in terms ofQ, we can differentiate (Doob’s h-transform) with respect to x to get: ∇h(x,t) =∇ Z R d Q T|t (X T = x T |X t = x)h(X T ,T )dx T = Z R d h(X T ,T )∇Q T|t (X T = x T |X t = x)dx T (4.87) To get the expression for the gradient of the logarithm, we divide both sides by h(x,t) to get: ∇ logh(x,t) = Z R d h(X T ,T ) h(x,t) ∇Q T|t (X T = x T |X t = x)dx T (4.88) Since we want to obtain a score-like function for the reference process, we can rewrite∇Q T|t = Q T|t ∇ log Q T|t then substitute the identity h(x,T ) = dΠ T|0 dQ T|0 (x|x 0 ) from (4.85) to get: ∇ logh(x,t) = Z R d h(X T ,T )Q T|t (x T |x) h(x,t) ∇ log Q T|t (x T |x)dx T = Z R d Π T|0 (x T |x 0 )Q T|t (x T |x) Q T|0 (x T |x 0 )h(x,t) ∇ log Q T|t (x T |x)dx T (4.89) The ratio Q T|t (x T |x) Q T|0 (x T |x 0 ) between reference conditionals can be rewritten using the Markov property of the reference measure which states that the distribution ofx,x T givenx 0 is equal, regardless of the order of sampling 0→ T → t or 0→ t→ T : Q T|0 (x T |x 0 )Q t|0,T (x|x 0 ,x T ) = Q t|0 (x|x 0 )Q T|t (x T |x) =⇒ Q T|t (x T |x) Q T|0 (x T |x 0 ) = Q t|0,T (x|x 0 ,x T ) Q t|0 (x|x 0 ) (4.90) Substituting the expression into (4.89), we get: ∇ logh(x,t) = Z R d Π T|0 (x T |x 0 )Q t|0,T (x|x 0 ,x T ) Q t|0 (x|x 0 )h(x,t) ∇ log Q T|t (x T |x)dx T (4.91) Now, we observe that the numerator Π T|0 (x T |x 0 )Q t|0,T (x|x 0 ,x T ) is equivalent to the joint law of (x,x T ) given X 0 = x 0 : Π T|0 (x T |x 0 )Q t|0,T (x|x 0 ,x T ) = Π t,T|0 (x,x T |x 0 )(Numerator of (4.91)) Furthermore, marginalizing the conditional bridge law Π(X 0:T |X 0 =x 0 ) =h(X T ,T)Q(X 0:T |X 0 = x 0 ) in (4.85) to time t gives us the following expression for the denominator: Q t|0 (x|x 0 )h(x,t) = Π t|0 (x|x 0 )(Denominator of (4.91)) by the definition of (Doob’sh-transform) ash(x,t) =E Q [h(X T ,T)|X t =x,X 0 =x 0 ]. Plugging these expressions into (4.91) yields: ∇ logh(x,t) = Z R d Π t,T|0 (x,x T |x 0 ) Π t|0 (x|x 0 ) |z =Π T|0,t (x T |x,x 0 ∇ log Q T|t (x T |x)dx T (Bayes’ rule) = Z R d ∇ log Q T|t (x T |x)dΠ T|0,t (x T |x,x 0 ) |z expectation over Π given X 0 = x 0 and X t = x = E Π T|t,0 ∇ log Q T|t (x T |x) X 0 = x 0 ,X t = x (4.92) 101 which can be used to express the correction drift in (4.86) as a conditional expectation of endpoint- conditioned drifts. Substituting this expression into the SDE for Π t|0 defined in (4.86), we get an SDE with non-Markov drift: dX t = f (X t ,t) + σ 2 t E Π T|t,0 ∇ log Q T|t (X T |X t ) X t ,X 0 dt + σ t dB t (4.93) Step 2: Show that Markovian Projection is the Minimizer of Reverse KL Divergence. Let Mbe an arbitrary Markov path measure that evolves via the SDEdX t = [f(X t ,t)+u(X t ,t)]dt+σ t dB t . Then, we can write the KL divergence between Π andMwith the same diffusion coefficientσ t and using Corollary 2.21 as the squared difference between their drifts: KL(Π∥M) = 1 2 Z T 0 E Π 0,t ∥σ t E Π T|0,t ∇ log Q T|t (X T |X t ) X t ,X 0 −u(X t ,t)∥ 2 dt(4.94) Minimizing this KL yields the optimalu ⋆ (x,t) that best approximates the non-Markovian drift σ t E Π T|t,0 ∇ log Q T|t (x T |x) X 0 ,X t using onlyX t across all time stepst. For any random variable Z, the minimizer of the expected squared loss with respect to a function is equal to the expectation of the random variable given only the input to the function: arg min b(X t ) E[∥Z−b(X t )∥ 2 ] = E[Z|X t ](4.95) Therefore, the minimizer is the optimal Markovian drift given by: u ⋆ (x,t) = σ t E Π T|t ∇ log Q T|t (X T |X t ) X t = x (4.96) which concludes the proof of the first condition (i) in Proposition 4.10. Step 3: Prove Equality of the Marginal Distributions.To prove the second condition (i) which states thatM ⋆ t = Π t for allt ∈[0,T], we first establish thatM ⋆ t and Π t solve the same Fokker-Planck equation given by: ∂ t p t (X t ,t) =−∇· ((f (X t ,t) +u ⋆ (X t ,t))p t (X t )) + σ 2 t 2 ∆p t (X t )(Fokker-Planck Equation) Since the divergence extracts only the drift that depends onX t , and the drift for Π contains only an additional dependence onX 0 , the effective drift in the (Fokker-Planck Equation) is equal for Π t and M ⋆ t . Given that both diffusion coefficients are also equal, we conclude that they both satisfy the same (Fokker-Planck Equation). Following the uniqueness of the solution to the Fokker-Planck equation given fixed initial conditions 15 , we conclude that the time marginals are equal M ⋆ t = Π t . □ While the Markovian projection provides a way of simulating endpoint-conditioned bridges with only dependence on the current state, it generally fails to preserve the bridge measure ofQ ·|0,T ≡ Q(·|x 0 ,x T ), which is the distribution over bridge paths conditioned on a pair of endpoints (x 0 ,x T )∼ Π 0,T . To define a measure that exactly matches the bridge of the endpoint-conditioned reference measureQ ·|0,T , we define the reciprocal projection which projects any path measure to the reciprocal class R(Q) of Q. To understand the reciprocal projection, we first define a reciprocal process and a unique property that will become useful in later sections. Proposition 4.11 (Reciprocal Processes (Lemma 1.4 in Jamison (1974))). A stochastic process (X t ) t∈[0,T ] on a measurable state spaceXis considered a reciprocal process if for any 0≤ s < t < τ ≤ Tand any bounded measurable functionφ(x) :X → R, the reciprocal 15 For proof, see (Bogachev et al., 2021) 102 property holds: E[φ(X t )|X 0:s ,X τ :T ] = E[φ(X t )|X s ,X τ ](Reciprocal Property) which means that the interior of any interval (s,τ) is conditionally independent of the rest of the trajectory given the two boundary states (X s ,X τ ). While reciprocal processes are generally not Markov, fixing either boundary state (X s orX τ ) to a constant yields a Markov process. Proof. To prove the Markov property of reciprocal processes given a fixed constant endpoint, we first observe that conditioning on a fixed constant does not change the conditional expectation. Suppose first thatX τ = ̄ xalmost surely. Lets≤ t 1 <· < t k < t < τandφ(x) :X → Rbe any bounded and measurable test function. SinceX τ is a constant, conditioning onX τ does not change the conditional expectation: E[φ(X t )|X t 1 ,...,X t k ] = E[φ(X t )|X t 1 ,...,X t k ,X τ ].(4.97) By the reciprocal property, once the two boundary states (X t k ,X τ ) are given, the interior of the interval (t k ,τ ) is conditionally independent of the past before t n . Therefore, we have: E[φ(X t )|X t 1 ,...,X t k ,X τ ] = E[φ(X t )|X t k ,X τ ].(4.98) Since X τ = ̄ x is constant, this reduces to: E[φ(X t )|X t k ,X τ ] = E[φ(X t )|X t k ].(4.99) Therefore, we conclude that the future stateX t can be determined with only knowledge of the current state X t k and independent of the past trajectory prior to t k : E[φ(X t )|X t 1 ,...,X t k ] = E[φ(X t )|X t k ],(4.100) which is exactly the Markov property. If insteadX s is fixed, the same conclusion follows by time reversal, since both the reciprocal and Markov properties are preserved under reversing time. □ Since we have shown that reciprocal processes are characterized by their conditional indepen- dence structure, we will now apply this idea to level of path measures. In particular, reciprocal processes naturally arise as mixtures of Markov bridges sharing the same conditional dynamics between endpoints. This observation leads to the notion of the reciprocal classR(Q) of a reference path measure Q. Intuitively, all path measures in the reciprocal classR(Q) generate trajectories with the same bridge dynamics asQ, differing only in how probability mass is assigned to the endpoint pairs (x 0 ,x T ). Concretely, this class consists of all path measures obtained by reweighting the endpoint distribution of Q while preserving its conditional bridge law. Definition 4.12 (Reciprocal Class). The reciprocal class ofQis the collection of all path measures that share the same conditional bridge distribution given fixed endpoints (x 0 ,x T )∼Π 0,T but may differ in their endpoint law, which defines how the endpoints are weighted. Formally, we define the reciprocal class R(Q) as: R(Q) :=Π∈P(C([0,T ]; R d )) : Π = Π 0,T Q ·|0,T (4.101) where Π 0,T ∈ P(R d , R d ) denotes an arbitrary endpoint coupling andQ ·|0,T is the endpoint- conditioned bridge distribution ofQ. Equivalently, a path measure belongs in the reciprocal 103 class Π∈R(Q) if it admits the mixture-of-bridges representation: Π(·) = Z R d ×R d Q ·|0,T (·|x 0 ,x T )dΠ 0,T (x 0 ,x T )(4.102) which can be written equivalently as Π = Π 0,T Q ·|0,T . Having characterized the reciprocal classR(Q) as the family of path measures that share the same bridge dynamics as the reference processQ, we now consider how to approximate an arbitrary path measurePby an element of the reciprocal class. A natural approach is to choose the measure inR(Q) that is closest toPin relative entropy. This is exactly the goal of the reciprocal projection, which we show is the unique minimizer of the KL divergence from P. Proposition 4.13 (Reciprocal Projection (Proposition 4 in Shi et al. (2023))). LetQbe a reference path measure andPbe an arbitrary path measure that we wish to project onto the reciprocal class ofQ. The reciprocal projection ofPonto the reciprocal classR(Q) is defined as: Π ⋆ := proj R(Q) (P)∈R(Q)(Reciprocal Projection) Then, Π ⋆ is the element of the reciprocal class which minimizes the KL divergence fromP: Π ⋆ = arg min Π∈R(Q) KL(P∥Π)(4.103) Furthermore, the reciprocal projection admits the mixture-of-bridges representation: Π ⋆ (X 0:T ) = Z R d ×R d Q ·|0,T (X 0:T |x 0 ,x T )dP 0,T (x 0 ,x T ) ⇐⇒ Π ⋆ = P 0,T Q ·|0,T (4.104) whereP 0,T denotes the endpoint distribution ofPandQ ·|0,T (·|x 0 ,x T ) is the reference bridge conditioned on the endpoints. Proof. We start by applying the (KL Divergence Chain Rule) from Lemma 1.4 to decomposeKL(P∥Π) into the sum of the divergence between endpoints and the divergence in bridge measure: KL(P∥Π) = KL(P 0,T ∥Π 0,T ) |z endpoint divergence + E (X 0 ,X T )∼P 0,T KL(P ·|0,T (·|X 0 ,X T )∥Π ·|0,T (·|X 0 ,X T )) | z bridge divergence (4.105) Since all elements of the reciprocal classR(Q) have the same bridge asQsuch that Π ·|0,T (·|x 0 ,x T ) = Q ·|0,T (·|x 0 ,x T ), we can write: KL(P∥Π) = KL(P 0,T ∥Π 0,T ) + E (X 0 ,X T )∼P 0,T KL(P ·|0,T (·|X 0 ,X T )∥Q ·|0,T (·|X 0 ,X T )) | z not dependent on Π (4.106) Since the bridge divergence is no longer dependent on the measure Π which we are optimizing, the only term left to minimize is the endpoint divergenceKL(P 0,T ∥Π 0,T ), which is minimized uniquely when: Π ⋆ 0,T = P 0,T (4.107) 104 which follows the strict convexity of the KL divergence. Given that Π ⋆ ∈R(Q) and Π ⋆ 0,T =P 0,T , we can write: Π ⋆ (·) = Z R d ×R d Q ·|0,T (·|x 0 ,x T )dP 0,T (x 0 ,x T )(4.108) which can equivalently be written as Π ⋆ =P 0,T Q ·|0,T and is exactly our definition of the (Reciprocal Projection).□ Now that we have shown that any path measure can be projected onto the reciprocal classR(Q) of a bridge measureQsuch that the bridge is preserved, we will show that there exists a unique Markov measure in the reciprocal class R(Q) which is also the solution to the (2.2). Proposition 4.14 (Solution to Dynamic Schr ̈odinger Bridge). The Markov measure in the reciprocal class ofQ, i.e.,M∈R(Q), that satisfiesM 0 =π 0 andM T =π T is the unique solution to the Schr ̈odinger bridge M = P ⋆ . Proof. By definition of the reciprocal classR(Q),MandQand share the exact same bridge such that M t|0,T =Q t|0,T for allt∈[0,T]. Thus, the Radon-Nikodym derivative dM dQ (X 0:T ) depends only on the endpoints (X 0 ,X T ) and can be written as: dM dQ (X 0:T ) = ξ(X 0 ,X T )(4.109) whereξ:R d × R d → Ris a measurable function. Since we definePto be Markov and the reference measureQis Markov by construction, conditioning on some intermediate stateX t yields the factorization: dM(X 0:T |X t ) dQ(X 0:T |X t ) = dM(X 0:t ,X t:T |X t ) dQ(X 0:t ,X t:T |X t ) = dM(X 0:t |X t ) dQ(X 0:t |X t ) dM(X t:T |X t ) dQ(X t:T |X t ) (4.110) where the past and future givenX t are independent. Given that the RND depends only on the endpoints via the measurable functionξ(X 0 ,X T ), there exist two measurable functionsa,b:R d → R that depend only on each endpoint, respectively, such that: dM dQ (X 0:T ) = ξ(X 0 ,X T ) = a(X 0 )b(X T )(4.111) Recall from Section 1 that the optimal endpoint lawπ ⋆ 0,T that solves the static SB problem factorizes into Schr ̈odinger potentials (φ, ˆφ) that are unique up to a constant. Therefore, we conclude: dM dQ (X 0:T ) = dπ ⋆ 0,T dq (X 0 ,X T ) = e φ(X 0 ) e ˆφ(X T ) (4.112) and we have shown that the Markov measure in the reciprocal classM∈R(Q) is unique and equals the Schr ̈odinger bridge M = P ⋆ .□ Building on the ideas of Markovian and reciprocal projections, we can define the Iterative Markovian Fitting (IMF) scheme (Shi et al., 2023) which constructs the optimal SB via alternating Markovian and reciprocal projections of an SDE which becomes the foundation of the practical DSBM algorithm which fits a parameterized velocity field by matching the control drift. Iterative Markovian Fitting (IMF) generates a sequence of path measures (P n ) n∈N with alter- nating Markovian projections and reciprocal projections defined as: P 2n+1 = proj M (P 2n ), P 2n+2 = proj R(Q) (P 2n+1 )(4.113) where the first path measure in the sequenceP 0 is the path measure in the reciprocal class of the reference measureP 0 ∈R(Q) that satisfies the marginal constraintsP 0 0 =π 0 andP 0 T =π T . The IMF procedure is grounded in three key theoretical results which are summarized as follows: 105 Figure 8: Iterative Markovian Fitting (IMF) vs. Iterative Proportional Fitting (IPF). While IPF alternates projections onto the marginal constraint sets Π(π 0 ,·) and Π(·,π T ), intermediate iterations generally do not preserve the endpoint coupling. In contrast, IMF performs Markovian and reciprocal projections that maintain both marginal constraintsP n ∈Π(π 0 ,π T ) at every iteration, progressively refining the path measure until convergence to the optimal bridge measure P ⋆ . (i)Pythagorean Identity (Lemma 4.15): The KL divergence between a mixture of bridges Π∈R(Q) in the recprocal class ofQand an arbitrary Markov measureM∈Msatisfies the Pythagorean identity: KL(Π∥M) =KL(Π|proj M (Π)) | z distance to Markovian projection +KL(proj M (Π)∥M) |z remaining distance to target (4.114) Similarly, the KL divergence between a path measurePand its reciprocal projectionproj R(Q) (P) satisfies the Pythagorean identity defined as: KL(P∥Π) =KL(P∥proj R(Q) (P)) |z distance to reciprocal projection + KL(proj R(Q) (P)∥Π) |z remaining distance to target (4.115) (i) Monotone Improvement (Proposition 4.16): The distance to the optimal SB measureP ⋆ is strictly decreasing with each iteration of IMF, such that: KL P n+1 ∥P ⋆ ≤ KL (P n ∥P ⋆ ),lim n→+∞ KL P n ∥P n+1 = 0(4.116) (i) Convergence Guarantee (Theroem 4.17): The sequence of path measures generated via IMF (P n ) n∈N converges to a unique fixed point P ⋆ that is exactly the SB measure P ⋆ : lim n→+∞ KL (P n ∥P ⋆ ) = 0 where P ⋆ = P ⋆ (4.117) To understand why each of these statements holds and build our intuition on the structure of Markovian and reciprocal projections, we will prove each of them and conclude with the convergence of the IMF procedure. Lemma 4.15 (Pythagorean Theorem of Markovian and Reciprocal Projections (Lemma 6 in Shi et al. (2023))). Let Π∈ R(Q) be a bridge measure in the reciprocal class ofQand proj M (Π) be the Markovian projection of Π. Given some arbitrary Markov measure that has 106 finite KL divergence with Π and proj M (Π), the following identity holds: KL(Π∥M) = KL(Π∥proj M (Π)) + KL(proj M (Π)∥M)(Markovian Projection Identity) which means that the KL divergence between Π andMis the sum of the KL incurred from projecting Π to the Markov class and the KL incurred from projectingproj M (Π) to the Markov measureM. Similarly, for any arbitrary path measureP∈P(C([0,T];R d )) projected onto the reciprocal class proj R(Q) (P), the KL divergence with Π∈R(Q) can be decomposed as: KL(P∥Π) = KL(P∥proj R(Q) (P)) + KL(proj R(Q) (P)∥Π)(Reciprocal Projection Identity) Proof. Step 1: Derive the Markovian Projection Identity. Intuitively, this step aims to derive the KL divergence to the Markov measure, which can be interpreted as the information lost when going from conditioning on (X 0 ,X t ) in Π to conditioning only onX t inM. To break this down, we define the key component that appears in both the drift of the Markovian projection and the mixture of bridges Π as: G t :=∇ log Q T|t (X T |X t )(4.118) Then, rewriting (Markovian Projection Drift) and the drift of the reciprocal process with additional conditioning on X 0 in terms of G t , we have: A t := E Π [G t |X 0 ,X t ] | z drift of reciprocal measure Π , B t := E Π [G t |X t ] |z drift of Markovian measure M (4.119) Using these definitions, we can rewrite the Markovian projection identity in terms of the integral differences between drifts as derived in Section 2.6. First, recall the identity: KL(Π∥M) |z (⋆) = KL(Π∥proj M (Π)) | z (⋄) + KL(proj M (Π)∥M) | z (♦) (4.120) For (⋆) defined as the KL divergence between Π∈R(Q) with control driftσ t A t and the Markov measure M with control drift u, we can write as the integral difference: KL(Π∥M) = 1 2 Z T 0 E Π 0,t ∥u(X t ,t)− σ t A t ∥ 2 dt(4.121) For (⋄), the Markovian projectionproj M (Π) has control driftσ t B t which removes conditioning on X 0 and yields following KL divergence with Π: KL(Π∥proj M (Π)) = 1 2 Z T 0 E Π 0,t ∥σ t A t − σ t B t ∥ 2 dt(4.122) Finally, for (♦), the KL divergence between the Markovian projectionproj M (Π) has control drift σ t B t and the Markov measure M with drift u(X t ,t) is given by: KL(proj M (Π)∥M) = 1 2 Z T 0 E Π 0,t ∥u(X t ,t)− σ t B t ∥ 2 dt(4.123) 107 Now, summing (⋄) and (♦) from the left-hand side of (4.120) and observing that projecting onto the subspace spanned by B t gives ⟨A t ,B t ⟩ =∥B t ∥ 2 and B t = E Π [A t |X t ], we have: KL(Π∥proj M (Π)) |z (⋄) + KL(proj M (Π)∥M) |z (♦) = 1 2 Z T 0 E Π 0,t ∥σ t A t − σ t B t ∥ 2 dt + 1 2 Z T 0 E Π 0,t ∥u(X t ,t)− σ t B t ∥ 2 dt = 1 2 Z E Π 0,t σ 2 t ∥A t ∥ 2 − 2σ 2 t ⟨A t ,B t ⟩ | z =2σ 2 t ∥B t ∥ 2 +σ 2 t ∥B t ∥ 2 +∥u(X t ,t)∥ 2 − 2σ t ⟨u(X t ,t),B t ⟩ + σ 2 t ∥B t ∥ 2 = 1 2 Z E Π 0,t σ 2 t ∥A t ∥ 2 −2σ 2 t ∥B t ∥ 2 + σ 2 t ∥B t ∥ 2 +∥u(X t ,t)∥ 2 − 2σ t ⟨u(X t ,t),B t ⟩ + σ 2 t ∥B t ∥ 2 = 1 2 Z E Π 0,t σ 2 t ∥A t ∥ 2 +∥u(X t ,t)∥ 2 − 2σ t ⟨u(X t ,t),B t ⟩ |z =⟨u(X t ,t),A t ⟩ = 1 2 Z E Π 0,t σ 2 t ∥A t ∥ 2 +∥u(X t ,t)∥ 2 − 2σ t ⟨u(X t ,t),A t ⟩ (given E Π t [⟨u,B t ⟩] = E Π 0,t [⟨u,A t ⟩]) = 1 2 Z T 0 E Π 0,t ∥u(X t ,t)− σ t A t ∥ 2 dt = KL(Π∥M) (4.124) where the final equality exactly yields the KL divergence betweenKL(Π∥M), which proves (Markovian Projection Identity). Next, we prove the (Reciprocal Projection Identity) with fewer steps. Step 2: Derive the Reciprocal Projection Identity.Let Π ⋆ =proj R(Q) (P) denote the reciprocal projection ofP. By the definition of reciprocal projection, Π ⋆ has the same endpoint law as P and the same conditional bridge as Q, defined as: Π ⋆ = Q ·|0,T P 0,T (4.125) Applying the (KL Divergence Chain Rule), we have: KL(P∥Π) = E P log dP dΠ = E P log dP dΠ ⋆ + log dΠ ⋆ dΠ = E P log dP dΠ ⋆ + E P log dΠ ⋆ dΠ = KL(P∥Π ⋆ ) + Z log dΠ ⋆ dΠ dP(4.126) Now, we leverage the key property of measures in the same reciprocal classR(Q) that share the same conditional bridge asQ. Therefore, the log RND depends only on the endpoints (X 0 ,X T ). In addition, the original path measurePand its reciprocal projection Π ⋆ share the same endpoint law, so we can write (4.126) as: KL(P∥Π) = KL(P∥Π ⋆ ) + Z log dΠ ⋆ dΠ (X 0:T ) |z depends on (X 0 ,X T ) only dP(X 0:T ) = KL(P∥Π ⋆ ) + Z log dΠ ⋆ 0,T dΠ 0,T (X 0 ,X T )dP(X 0 ,X T ) | z =dΠ ⋆ 0,T = KL(P∥Π ⋆ ) + Z log dΠ ⋆ 0,T dΠ 0,T (X 0 ,X T )dΠ ⋆ 0,T (X 0 ,X T ) |z =KL(Π ⋆ ∥Π) = KL(P∥Π ⋆ ) + KL(Π ⋆ ∥Π)(4.127) which recovers the (Reciprocal Projection Identity) by subsituting back Π ⋆ = proj R(Q) (P). □ Using Lemma 4.15, we can prove that iterating between Markov and reciprocal projections yields a monotonically decreasing KL divergence to the Schr ̈odinger bridge measure, which will be the foundation for our final convergence proof. Proposition 4.16 (Monotonically Decreasing KL Divergence (Proposition 7 in Shi et al. (2023))). Given a sequence of Markov and reciprocal projections (P n ) n∈N and the Schr ̈odinger 108 bridge path measureP ⋆ , the reverse KL divergences betweenP n andP ⋆ decreases monotoni- cally: ∀n∈ N,KL(P n+1 ∥P ⋆ )≤ KL(P n ∥P ⋆ )≤∞(4.128) and in the limitn→∞, the KL divergence between subsequent projections converges to zero: lim n→∞ KL(P n ∥P n+1 ) = 0(4.129) Proof. Recall that the Schr ̈odinger bridge is the unique path measure that is MarkovP ⋆ ∈Mand in the reciprocal classP ⋆ ∈R(Q). Therefore, both (Markovian Projection Identity) and (Reciprocal Projection Identity) from Lemma 4.15 hold with respect toP ⋆ . This means that for allP n , the following identity holds: KL(P n ∥P ⋆ ) = KL(P n ∥P n+1 ) |z ≥0 +KL(P n+1 ∥P ⋆ ) =⇒ KL(P n ∥P ⋆ )≥ KL(P n+1 ∥P ⋆ ) (4.130) whereP n+1 is either the Markovian projection ofP n ifP n was generated from a reciprocal projection or the reciprocal projection ofP n+1 ifP n was generated from a Markovian projection. This proves the first part of the proposition. Next, we show that at the limit asn→∞, the KL between iterations converges to zero. To do this, we can compute the accumulated KL divergence overNiterations using the same identity from (4.130). To do this, we apply the identity for each iteration n = 0,...,N to get: n = 0 : KL(P 0 ∥P ⋆ ) = KL(P 0 ∥P 1 ) + KL(P 1 ∥P ⋆ ) n = 1 : KL(P 1 ∥P ⋆ ) = KL(P 1 ∥P 2 ) + KL(P 2 ∥P ⋆ ) . . . n = N : KL(P N ∥P ⋆ ) = KL(P N ∥P N +1 ) + KL(P N +1 ∥P ⋆ )(4.131) Observing that the intermediate KL divergences withP ⋆ (indicated in blue) cancel after summing over n = 0,...,N (telescoping identity), we can write: KL(P 0 ∥P ⋆ ) |z fixed ≤∞ = N X n=0 KL(P n ∥P n+1 ) + KL(P N +1 ∥P ⋆ ) =⇒ N X n=0 KL(P n ∥P n+1 )≤∞(4.132) where we observe that the KL divergence between the initial path measureP 0 andP ⋆ is fixed and bounded. Since (4.132) holds as we increaseNto infinity, we have shown that the series of non-negative KL divergences is boundedKL(P n ∥P n+1 ) is finite and bounded, the additive terms must converge to zero: ∀n∈ N, N X n=0 KL(P n ∥P n+1 )≤∞ =⇒lim n→∞ KL(P n ∥P n+1 ) = 0(4.133) and we conclude our proof.□ This leads to the final result, which ensures that an IMF sequence converges to the optimal SB path measure P ⋆ . Proposition 4.17 (Iterative Markovian Fitting Converges to the Unique Schr ̈odinger Bridge). The sequence of path measures (P n ) n∈N generated from alternating Markovian and reciprocal 109 projections of the IMF algorithm has a unique fixed pointP ⋆ which equals the Schr ̈odinger bridge. Furthermore, in the limit n→∞, the KL divergence converges to the fixed point: lim n→∞ KL(P n ∥P ⋆ ) = 0(4.134) Proof Sketch. By Proposition 4.16, we know that each path measure in the sequence (P n ) n∈N decreases the KL divergence toP ⋆ , so the sequence remains trapped in a compact region in path space that is bounded below by zero, since KL divergence is non-negative. Therefore, both the sequence of Markovian projections and reciprocal projections converge to their optimal fixed points M ⋆ ∈Mand Π ⋆ ∈R(Q). From Proposition 4.16, we also have that the the KL divergence between each iteration converges to zero, which implies that the fixed points of the Markovian and reciprocal projections coincide exactly 16 : lim n→∞ KL(P n ∥P n+1 ) = 0 =⇒ KL(M ⋆ ∥Π) = 0 =⇒ M ⋆ = Π ⋆ = P ⋆ (4.135) which means that the shared limit of the Markov and reciprocal projections is both Markov and in the reciprocal classR(Q), and therefore, must be the Schr ̈odinger bridgeP ⋆ . Given that both subsequences converge toP ⋆ , we have that the full sequence (P n ) n∈N also converges toP ⋆ and the KL divergence converges to zero. 17 □ The relationship between Markov and reciprocal projections reveals a key structural property of the Schr ̈odinger bridge. The (Markovian Projection) enforces the Markov property by selecting the closest Markov process in relative entropy to some bridge measure, and the (Reciprocal Projection) adjusts the path measure so that the endpoint marginals match the prescribed distributions, while preserving the bridge structure inherited from the reference process. Crucially, the Schr ̈odinger bridgeP ⋆ lies exactly at the equilibrium of these two constraints, as it is the unique path measure that simultaneously satisfies the endpoint conditions and remains the closest Markov measure to the reference bridge dynamics. We further show that alternating between performing Markovian and reciprocal projections yields a unique fixed point that coincides withP ⋆ , which is exactly what defines the Iterative Markovian Fitting (IMF) procedure. We will revisit the IMF procedure in Section 6.3, where we apply this procedure in the context of generative modeling. 4.6 Stochastic Interpolants to Schr ̈odinger Bridges The stochastic interpolants framework (Albergo et al., 2025) can be used to construct the Schr ̈odinger bridge solution. First, we will provide some background on the framework, which will naturally lead to its extension in solving the SB problem. Definition 4.18 (Stochastic Interpolant (Albergo et al., 2025)). Letπ 0 ,π T ∈P(R d ) be two probability densities on the state space. The stochastic interpolant betweenπ 0 andπ T is a stochastic process X 0:T of the form: x t = I(x 0 ,x T ,t) + γ(t)z, t∈ [0,T ](4.136) where the following are satisfied: 16 Note thatM ⋆ and Π ⋆ are used to denote the Markovian and reciprocal projections for any path measure, but they only coincide at the unique fixed point when they are equal to the Schr ̈odinger bridge. 17 for more rigorous proof, see (Shi et al., 2023) 110 (i)The mapI ∈ C 2 ((C 2 (R d × R d )) d ,[0,T]) has boundary conditionsI(x 0 ,x T ,0) =x 0 and I(x 0 ,x T ,T ) = x T and controlled time variation: ∃C 1 <∞ s.t. |∂ t I(x 0 ,x T ,t)|≤ C 1 |x T −x 0 |, ∀(x 0 ,x T ,t)∈ [0,T ]× R d × R d (4.137) (i)The scalar noise functionγ: [0,T]→ Rsatisfiesγ(0) =γ(1) = 0 andγ(t)>0 for all t∈ (0,T ). In addition, γ 2 ∈ C 2 ([0,T ]). (i)The pair (x 0 ,x T ) are sampled from a probability measureπ 0,T whose marginals areπ 0 and π T defined as: π 0,T (dx 0 , R d ) = π 0 (x 0 )dx 0 , π 0,T (R d ,dx T ) = π T (x T )dx T (4.138) (iv) The Gaussian random variable z ∼N (0,I d ) is independent of (x 0 ,x T ). Intuitively, the stochastic interpolant is a general method of connecting samples from two marginal distributions with a deterministic pathI(x 0 ,x T ,t) that is perturbed by a time-dependent Gaussian diffusionγ(t)zalong the interior of the time interval. By definition, the noiseγ(t)zvanishes att= 0 and t = T , ensuring that the process matches exactly the terminal marginals π 0 and π T . Theorem 4.19 (Properties of Stochastic Interpolants (Theorem 2.6 in Albergo et al. (2025))). The stochastic interpolant x t = I(x 0 ,x T ,t) satisfies the following properties: ∂ t p t +∇· (p t v) = 0(4.139) where the velocity is defined as the expectation of the time derivative: v(x,t) = E p t [ ̇ x t |x t = x] = E[∂ t I(x 0 ,x T ,t) + ̇γ(t)z|x t = x](4.140) which is bounded on the domain of the density function p t (x) given by: ∀t∈ [0,T ] : Z R d ∥v(x,t)∥ 2 p t (x)dx <∞(4.141) Proof. We start with the definition of the Fourier transformF(ω,t)[p t ] of the densityp t (x) for the random variable x t = I(x 0 ,x T ,t) + γ(t)z, given by: F (ω,t)[p t ] = Z R d e iω·x t p t (x) = E e iω·x t = E h e iω·(I(x 0 ,x T ,t)+γ(t)z) i (4.142) whereω ∈ R d is thed-dimensional Fourier frequency variable. To show thatp t (x) satisfies the continuity equation in (4.139), we can take the time derivative of its Fourier transformF(ω,t)[p t ] to get: ∂ t F (ω,t)[p t ] = ∂ t E h e iω·(I(x 0 ,x T ,t)+γ(t)z) i = E h ∂ t e iω·(I(x 0 ,x T ,t)+γ(t)z) i = E h e iω·x t ∂ t iω· (I(x 0 ,x T ,t) + γ(t)z) i = iω· E h e iω·x t (∂ t I(x 0 ,x T ,t) + ̇γ(t)z) i | z (⋆) (4.143) 111 Now, we can apply the law of total expectationE[X] =E[E[X|Y]] to write (⋆) with respect to a conditional expectation on X t = x: ∂ t F (ω,t)[p t ] = iω· E h e iω·x t (∂ t I(x 0 ,x T ,t) + ̇γ(t)z) i = iω· E h e iω·x E (∂ t I(x 0 ,x T ,t) + ̇γ(t)z)|X t = x i = iω· Z R d e iω·x E (∂ t I(x 0 ,x T ,t) + ̇γ(t)z)|X t = x | z =:v(x,t) p t (x)dx = iω· Z R d e iω·x v(x,t)p t (x)dx(4.144) Next, we show that the Fourier transform of−∇·(vp t ) is exactlyiωF(ω,t)[vp t ]. Given that ∇· (vp t ) = P d j=1 ∂ x j (vp t ) j , we have: F (ω,t)[−∇· (vp t )] =− d X j=1 Z R d e iω·x | z u ∂ x j (vp t ) j | z dv dx = h e iω·x (vp t ) i ∞ −∞ | z =0 + d X j=1 Z R d (vp t ) j ∂ x j e iω·x dx = d X j=1 Z R d (vp t ) j ∂ x j e i P d ℓ=1 ω ℓ x ℓ dx = d X j=1 Z R d (vp t ) j iω j e iω·x dx = d X j=1 iω j Z R d e iω·x (vp t ) j dx = iω· Z R d e iω·x vp t dx(4.145) which proves that v and p t satisfies the continuity equation in the real space: ∂ t F (ω,t)[p t ] =F (ω,t)[−∇· (vp t )] =⇒ ∂ t p t =−∇· (vp t ) (4.146) and we conclude our proof.□ Having established the fundamental properties of stochastic interpolants, we now observe that they induce a family of time-evolving densities (p t ) t∈[0,T ] governed by the continuity equation (4.139). This equation characterizes deterministic mass transport under a velocity fieldv, and forms the core dynamical constraint underlying optimal transport. In particular, stochastic interpolants provide a constructive way to define admissible trajectories that transport probability mass between endpoint distributions. However, while stochastic interpolants describe valid transport dynamics, they do not yet specify which trajectory is optimal. The SB problem resolves this ambiguity by selecting, among all admissible paths satisfying the same marginal constraints, the one that is closest to a reference stochastic process in the sense of minimizing path-space KL divergence. As we now show, this optimality can be expressed through stochastic control, where the velocity fieldvis parameterized via a controlled drift that minimizes a quadratic control cost while matching the prescribed marginals. 112 We begin by recalling the (Dynamic SB Problem), which aims to determine the optimal control drift and density evolution (u ⋆ ,p ⋆ t ) that solve the minimization problem: inf (u,p t ) Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x)dxdt s.t. ( ∂ t p t =−∇· (p t u) + ε∆p t p 0 = π 0 , p T = π T (4.147) which can be rewritten as solving (HJB-FP System) given by: ( ∂ t ψ t + 1 2 ∥∇ψ t ∥ 2 =−ε∆ψ t ∂ t p ⋆ t +∇· (p ⋆ t u ⋆ ) = ε∆p ⋆ t s.t. ( p ⋆ 0 = π 0 p ⋆ T = π T (4.148) where we simplify the setting by setting f ≡ 0 and ε = σ 2 t 2 . Before we construct the stochastic interpolant that solves the SB problem, we will define an invertible map that generates the optimal SB density p ⋆ t that solves (4.147). Definition 4.20 (Invertible Map). Define an invertible mapM:R d ×[0,T]→ R d where M,M −1 ∈ C 1 ([0,T ], (C d (R d )) d ) such that: p ⋆ t (x) = M (·,t) # N (0,I d )(4.149) In other words, given a Gaussian random variable z ∼N (0,I d ), we have x t = M (z,t)∼ p ⋆ t . Given the existence of the invertible map, we can now derive the stochastic interpolant that solves the SB problem. Lemma 4.21 (Stochastic Interpolant Form of SB Solution (Lemma 3.12 in Albergo et al. (2025)). Given the existence of an invertible mapMdefined in Definition 4.20, the optimal densityp ⋆ t that solves the dynamic SB problem can be written as a stochastic interpolant of the form: x t = M α(t)M −1 (x 0 , 0) + β(t)M −1 (x T ,T ),t + γ(t)(4.150) whereα 2 (t) +β 2 (t) +γ 2 (t) = 1. This corresponds to defining the interpolant function from Definition 4.18 as I(x 0 ,x T ,t) = M (α(t)M −1 (x 0 , 0) + β(t)M −1 (x T ,T )). Proof. Intuitively, this Lemma states that the density p ⋆ t that solves the HJB-FP system defined in (4.148) is exactly the distribution of the random variable obtained defined in (4.150). By definition of the mapM(·,t) which transports a standard Gaussian to the target densityp ⋆ t at timet, so the inverse mapM −1 (·,t) must transport the target densityp ⋆ t back to a standard Gaussian, such that: x 0 ∼ π 0 , M −1 (x 0 , 0)∼N (0,I d )(4.151) x T ∼ π T , M −1 (x T ,T )∼N (0,I d )(4.152) Sincez ∼N(0,I d ) is also sampled from a standard Gaussian andx 0 ,x T ,zare drawn independently in the stochastic interpolant construction, we have that the linear combination of independent standard Gaussians given by: α(t)M −1 (x 0 , 0) + β(t)M −1 (x T ,T ) + γ(t)z ∼N (0,α 2 (t)I d + β 2 (t)I d + γ 2 (t)I d ) =N (0,I d ) (4.153) 113 is a Gaussian with zero-mean and covariance (α 2 (t) +β 2 (t) +γ 2 (t))I d . Since we definedα 2 (t) + β 2 (t) +γ 2 (t) = 1, then this is just a standard GaussianN(0,I d ). Therefore, we can apply the definition of the mapM(·,t) on the linear combination (4.153) which yields a random variable from the target distribution p ⋆ t : x t = M α(t)M −1 (x 0 , 0) + β(t)M −1 (x T ,T ),t + γ(t)z ∼ p ⋆ t (4.154) which concludes the proof.□ Lemma 4.21 shows that the densityp ⋆ t that solves the HJB-FP system can be obtained as the distribution of a random variable constructed by interpolation of samples from the terminal marginals x 0 ∼ π 0 andx T ∼ π T in Gaussian space, which establishes the form of the optimal interpolantI ⋆ that generates the Schr ̈odinger bridge marginals. Now, we can derive an objective whose minimizer recovers this optimal interpolant I ⋆ . Proposition 4.22 (Solving SB problem with Stochastic Interpolants (Theorem 3.13 in Albergo et al. (2025)). Define a scalar functionγ(t) : [0,T]→[0,1) which returns zero at the terminal time points (i.e.,γ(0) =γ(T) = 0), returns non-zero at intermediate times (i.e., ∀t∈ (0,T ), γ(t) > 0), and satisfies γ ∈ C 2 ((0,T )) and γ 2 ∈ C 1 ([0,T ]). Then, given independentx 0 ,x T ,zsolving the max-min problem over ˆ I ∈ C 1 ([0,T],(C 1 (R d × R d )) d ) and ˆ u∈ C 0 ([0,T ], (C 1 (R d )) d ) given by: max ˆ I min ˆ u Z T 0 E 1 2 ∥ ˆ u(x,t)∥ 2 − ∂ t ˆ I(x 0 ,x T ,t) + ̇γ(t)− εγ −1 (t) z · ˆ u(x,t) dt(4.155) s.t. ( x t = ˆ I(t,x 0 ,x T ) + γ(t)z x 0 ∼ π 0 , x T ∼ π T , z ∼N (0,I d ) (4.156) where, given the existence of the invertible mapM, all optimal (I ⋆ ,u ⋆ ) produces the stochastic interpolantx t =I ⋆ (x 0 ,x T ,t) +γ(t)zwith marginalsp ⋆ t that satisfy the continuity equation of the form: ∂ t p t =−∇· ( ˆp t ˆ v)(4.157) where ˆ vis the effective velocity field that accounts for the Gaussian perturbation induced by γ(t)z. Proof. To prove this, we begin by defining the effective velocity ˆ vof the stochastic interpolant and rewriting the objective as an optimization over ˆ v. Then, we write the constrained objective using Lagrange multipliers and solve for the optimal solution. First, we rewrite the Fokker-Planck equation into a continuity constraint that must be satisfied by the effective velocity: ∂ t p t =−∇· ( ˆp t v) + ε∆ ˆp t =⇒ ∂ t p t =−∇· ( ˆp t v) + ε∇·∇ ˆp t =⇒ ∂ t p t =−∇· ( ˆp t (v− ε∇ log ˆp t ) |z =: ˆ v ) (4.158) To define the effective velocity that accounts for the contribution of the Gaussian perturbation, we need to incorporateε∇ log ˆp t . Since the stochastic interpolant is defined asx t =I(x 0 ,x T ,t) +γ(t)z, the conditional distribution ofx t given (x 0 ,x T ) is Gaussian with varianceγ 2 (t). A standard identity for Gaussian variables implies that E[z|X t = x] =−γ(t)∇ log ˆp t (x),(4.159) 114 where ˆp t is the density of x t . Therefore, we have −εγ −1 (t)E[z|X t = x] = ε∇ log ˆp t (x)(4.160) Now, we can define the average effective velocity as: ˆ v(x,t) := E[∂ t ˆ I(x 0 ,x T ,t) + ( ̇γ(t)−εγ −1 (t))z|X t = x](4.161) where−εγ −1 (t) corrects for the Gaussian perturabtion. Then, rewriting the expectation in the original objective as the integral, we have: max ˆ I min ˆ u Z T 0 E 1 2 ∥ ˆ u( ˆ x t ,t)∥ 2 − ∂ t ˆ I(x 0 ,x T ,t) + ̇γ(t)− εγ −1 (t) z · ˆ u( ˆ x t ,t) dt = max ˆp,v min ˆ u Z T 0 Z R d 1 2 ∥ ˆ u(x,t)∥ 2 − E h ∂ t ˆ I(x 0 ,x T ,t) + ̇γ(t)− εγ −1 (t) z X t = x i · ˆ u(x,t) ˆp t (x)dxdt = max ˆp,v min ˆ u Z T 0 Z R d 1 2 ∥ ˆ u(x,t)∥ 2 ˆp t (x)− ˆ v(x,t)· ˆ u(x,t) ˆp t (x)dxdt(4.162) s.t. ∂ t ˆp t +∇· (v ˆp t ) = ε∆ ˆp t ,ˆp 0 = π 0 ,ˆp T = π T Rewriting (4.162) as an unconstrained objective with Lagrange multipliersψ t (x),η 0 (x) andη T (x), we have: max ˆp, ˆ v min ˆ u Z T 0 Z R d 1 2 ∥ ˆ u(x,t)∥ 2 ˆp t (x)− ˆ v(x,t)· ˆ u(x,t) ˆp t (x)dxdt − Z T 0 Z R d ψ t (x) (∂ t ˆp t (x) +∇· ( ˆ v(x,t) ˆp t (x)))dxdt + Z R d η 0 (x) ( ˆp 0 (x)− π 0 (x))dt + Z R d η T (x) ( ˆp T (x)− π T (x))dt (4.163) The optimal (p ⋆ t ,u ⋆ ) that minimizes the Lagrangian pointwise satisfies the following optimality conditions: (i) Varying with respect to ψ t (x) enforces: ∂ t ˆp t +∇· ( ˆ v ˆp t ) = 0(4.164) (i) Varying with respect to ˆ u enforces: ∂ ∂u 1 2 ∥ ˆ u∥ 2 p t − ˆ u· ( ˆ v ˆp t ) = ˆ u ˆp t − ( ˆ v ˆp t ) = 0 =⇒ u ⋆ = ˆ v(4.165) (i) Since ( ˆ v ˆp t ) appears in − ˆ u· ( ˆ v ⋆ ˆp t ) and ψ t (x)∇· ( ˆ v ˆp t ), we can integrate by parts to get: − Z ˆ u· ( ˆ v ˆp t )− Z ψ t ∇· ( ˆ v ˆp t ) =− Z ˆ u· ( ˆ v ˆp t )− [ψ t (x)( ˆ v ˆp t )] ∞ −∞ | z =0 + Z ∇ψ t · ( ˆ v ˆp t ) =− Z ˆ u· ( ˆ v ˆp t ) + Z ∇ψ t · ( ˆ v ˆp t ) = Z (∇ψ t − ˆ u)· ( ˆ v ˆp t ) =⇒ ∇ψ t −u ⋆ = 0(4.166) 115 (iv)Isolating the terms that depend onˆp t (excluding those that involve ˆ v ˆp t , which we accounted for previously) and integrating by parts to factor out ˆp t , we have: Z T 0 Z R d 1 2 ∥ ˆ u∥ 2 ˆp t − ψ t ∂ t ˆp t dxdt = Z T 0 Z R d 1 2 ∥ ˆ u∥ 2 ˆp t + (∂ t ψ t ) ˆp t dxdt = Z T 0 Z R d 1 2 ∥ ˆ u∥ 2 + (∂ t ψ t ) |z vanishes pointwise at minimum ˆp t dxdt =⇒ ∂ t ψ t + 1 2 ∥u ⋆ ∥ 2 = 0(4.167) (v) Varying with respect to η 0 (x) and η T (x) yields p ⋆ 0 = π 0 and p ⋆ T = π T . Putting it all together, (i) and (i) enforceu ⋆ =∇ψ t , which can be substituted into (i) to get the following system: ( ∂ t ψ t + 1 2 ∥∇ψ t ∥ 2 = 0 ∂ t p ⋆ t +∇· (p ⋆ t ∇ψ t ) = 0 s.t. ( p 0 = π 0 p T = π T (4.168) which is exactly the HJB-FP system satisfied by the solution to the SB problem with given in (4.148) with vanishing diffusion, since the Gaussian diffusion is absorbed into the marginal densityp t through the effective velocity ˆ v . Given the existance of the invertible map from Definition 4.20 that yields the optimal densityp ⋆ t , we have that the pair (I ⋆ ,u ⋆ ) that solves the original max-min objective in (4.155) exists which we have shown yields the optimal densityx t ∼ p ⋆ t forx t =I ⋆ (x 0 ,x T ,t) +γ(t)z. □ This result shows that stochastic interpolants provide a alternative representation of the Schr ̈odinger bridge dynamics (u ⋆ ,p ⋆ t ), where the optimal control and velocity fields coincide and are given by the gradient of the Lagrange multiplier∇ψ t , yielding the coupled HJB-FP system that define the optimality conditions of the SB solution. Therefore, sampling from the stochastic interpolant x t =I ⋆ (x 0 ,x T ,t) +γ(t)z, wherez ∼ N(0,I d ) generates trajectories whose marginals match the marginal density flow p ⋆ t of the SB path measure P ⋆ . 4.7 Closing Remarks for Section 4 This section explored the theoretical foundations for building a Schr ̈odinger bridge using several approaches. While each approach originates from a different mathematical viewpoint, they ultimately converge to a unified form of a Markov control drift that minimally corrects the uncontrolled reference dynamics such that they reconstruct the prescribed marginal distributions. The key takeaway of this section is that stochastic bridges are not arbitrary conditioned processes, but rather minimal-entropy corrections of a reference diffusion that preserve the bridge structure while introducing the smallest possible dynamical adjustment. This adjustment consistently appears as a gradient of a logarithmic potential, providing a unifying perspective to the structure of the Schr ̈odinger bridge. The different constructions introduced in this section can therefore be understood as alternative ways of identifying the optimal control drift that modifies the reference dynamics: (i) Mixture of Conditional Bridges (Section 4.1): This section shows that the dynamic SB P ⋆ can be expressed as a mixture of endpoint-conditioned stochastic bridges given samples from the optimal endpoint law (x 0 ,x T )∼ π ⋆ 0,T , which can be constructed with a conditional driftu(x,t;x 0 ,x T ). This decomposition separates the SB problem into estimating the optimal static coupling and learning the conditional bridge dynamics. 116 (i)Time-reversal formula (Section 4.2): This section shows that the time-reversal formula yields a backward correction term∇ logp T−s (x) that corresponds to the score function in score-based generative modeling. This formulation models the endpoint-conditioned bridge for uncontrolled forward process. (i)Forward-backward SDEs (Section 4.3): This section analyzes how the theory of for- ward–backward theory generalizes the time-reversal formula for a forward process containing a non-deterministic control drift∇ logφ t (x) which yields a backward control drift∇ log ˆφ t (x) that evolves via a system of forward-backward SDEs. (iv)Doob’sh-transform (Section 4.4): This section defines anh-functionh(x,t) that reweights path transitions by its potential at timeτ. The reweighted path measureP h is defined by an SDE with a correction term∇ logh(x,t). By definingh(x,t) :=E Q [φ T (X T )|X t =x], we show that the tilted path measure recovers the Schr ̈odinger bridge P ⋆ . (v)Markov and reciprocal projections (Section 4.5): This section shows that the optimal Schr ̈odinger bridgeP ⋆ can be interpreted as the equilibrium point between projections onto the space of Markov path measures and the reciprocal classR(Q) containing mixtures of bridges under the reference pathQ. In this formulation, the optimal drift is expressed as an expectation over the target-conditioned path measure E Π T|t [∇ log Q T|t (X T |X t )|X t ]. (vi)Stochastic interpolants (Section 4.6): The stochastic interpolant framework represents the bridge by expressing the intermediate state asx t =I ⋆ (x 0 ,x T ,t) +γ(t)z, whereI ⋆ is the optimal interpolant between samplesx 0 ∼ π 0 andx T ∼ π T andz ∼ N(0,I d ) is Gaussian noise. We show that the induced velocity field satisfies the same optimality conditions that characterize the Schr ̈odinger bridge dynamics. While this section provides a principled framework for constructing stochastic bridges between prescribed endpoint distributions that solve the (Dynamic SB Problem), recent advances in generative modeling have motivated a variety of specialized Schr ̈odinger bridge formulations tailored to different modeling assumptions and problem settings. These variants extend the original framework in several directions, ranging from alternative reference processes to mean-field interactions, unbalanced mass transport, and multi-marginal and multi-modal constraints. In the next section, we analyze several of these problem variations and discuss how they modify our dynamic SB problem formulation while preserving its core principle: constructing stochastic dynamics that minimally deviate from a reference process while matching prescribed marginal distributions. 117 5. Variations of the Schr ̈odinger Bridge Problem In previous sections, we have established the foundational theories and intuition behind the classical static and dynamic Schr ̈odinger bridge problem and have shown how to derive stochastic bridges from scratch using various techniques. Now, we are ready to describe diverse variations of the SB problem that have been introduced in conjunction to novel generative modeling techniques, each of which are specialized for different settings and tasks. Specifically, we analyze the Gaussian SB problem (Section 5.1), the generalized SB problem (Section 5.2), the multi-marginal SB problem (Section 5.3), the unbalanced SB problem (Section 5.4), the branched SB problem (Section 5.5), and finally the fractional SB problem (Section 5.6). For each problem variation, we provide the intuition and formal definition of the problem, the necessary theoretical background, and relevant proofs and derivations that are useful for interpreting the problem and its optimal solution. 5.1 Gaussian Schr ̈odinger Bridge Problem While the (Dynamic SB Problem) defined in Section 2 does not admit a close form solution in general, in the special case where the marginal distributionsπ 0 ,π T are Gaussian distributions defined asπ 0 ∼ N(μ 0 ,Σ 0 ),π T ∼ N(μ T ,Σ T ), the SB solution can be solved in closed form (Bunne et al., 2023). This special case of the dynamic SB problem is called the Gaussian Schr ̈odinger Bridge (SB) problem (Bunne et al., 2023; Mallasto et al., 2022). Beforing defining the Gaussian SB problem, we start by defining the Gaussian formulation of the entropic OT problem (Bojilov et al., 2016; Janati et al., 2020; Mallasto et al., 2022), which will become crucial for our later derivation of the closed-form solution in the Gaussian SB setting. Lemma 5.1 (Static Entropy-Regularized Gaussian Optimal Transport). Letπ 0 =N(μ,Σ) andπ T =N(μ,Σ) be Gaussian probability measures onR d whereμ,μ ′ ∈ R d are the mean vectors and Σ. Consider the entropy-regularized optimal transport problem min π 0,T ∈Π(π 0 ,π T ) Z ∥x T −x 0 ∥ 2 dπ(x 0 ,x T ) + 2σ 2 KL (π 0,T ∥π 0 ⊗ π T )(5.1) whereσ ≥0 and Π(π 0 ,π T ) denotes the set of couplings with marginalsπ 0 andπ T . Then, the unique optimal coupling π ⋆ 0,T is Gaussian and satisfies π ⋆ 0,T ∼N " μ 0 μ T # , " Σ 0 C σ C ⊤ σ Σ T #! (5.2) where C σ := 1 2 Σ 1/2 0 D σ Σ −1/2 0 − σ 2 I d ,where D σ := 4Σ 1/2 0 Σ T Σ 1/2 0 + σ 4 I d 1/2 (5.3) In particular, whenσ= 0, the solution reduces to the classical Gaussian optimal transport coupling with quadratic transport cost. Intuition. While we omit the full proof 18 , we observe that when transporting between two Gaussian distributions under entropy regularization, the optimal coupling remains Gaussian and is fully characterized by how the two variables are correlated. Since the marginals are fixed to beπ 0 andπ T , the only degree of freedom is the cross-covariance C σ , which determines how samples are paired. 18 See Janati et al. (2020) for proof. 118 The objective balances two competing effects: the quadratic cost∥x T − x 0 ∥ 2 encourages pairs (x 0 ,x T ) to be as close as possible, whereas the entropy regularization termKL (π 0,T ∥π 0 ⊗ π T ) penalizes deviations from independence, pushing the coupling toward the product measure where x 0 andx T are uncorrelated. Therefore, the optimal cross-covarianceC σ can be interpreted as the optimal trade-off between these two objectives. Given this result, we are ready to define the Gaussian Schr ̈odinger Bridge (SB) problem and derive its closed form solution. Definition 5.2 (Gaussian Schr ̈odinger Bridge Problem). Letπ 0 =N(μ,Σ) andπ T =N(μ,Σ) be Gaussian probability measures onR d whereμ,μ ′ ∈ R d are the mean vectors and Σ,Σ ′ ∈ R d×d are the covariances, and letQbe a reference path measure. The Gaussian Schr ̈odinger bridge seeks the path measure that matches the Gaussian marginals while minimizing the relative entropy with respect to Q: P ⋆ =arg min P∈P(C([0,T ];R d )) KL(P∥Q) : π 0 =N (μ, Σ),π T =N (μ ′ , Σ ′ ) (Gaussian SB Problem) which can also be written in the form of a (Entropy-Regularized Dynamic OT) a as: inf (p t ,v) Z T 0 E p t 1 2 ∥v(x,t)∥ 2 + σ 4 t 8 ∥∇ logp t (x)∥ 2 dt s.t. ( ∂ t p t =−∇· (p t v) p 0 =N (μ, Σ), p T =N (μ ′ , Σ ′ ) (5.4) where we set f ≡ 0. a to align notation with (Bunne et al., 2023), we denote the full driftf N (x,t) without scaling byσ t , which results the additional factor of σ 2 t 4 in the Fisher information term. To obtain a tractable characterization of this problem, we exploit the special structure of Gaussian measures, which are uniquely defined by their mean and covariance. When the marginals remain Gaussian along the interpolation, the evolution of the process is fully characterized by the trajectories of its meanμ t ∈ R d and covariance Σ t ∈ R d×d . In this setting, the Gaussian Schr ̈odinger bridge can be interpreted as minimizing the energy of the change in covariance matrices as they move along the manifold of symmetric positive definite matrices Σ∈ S d ++ , whose tangent space is the space of symmetric matrices, whose tangent space is the space of symmetric matrices: T Σ S d ++ :=U ∈ R d×d : U ⊤ = U(5.5) The natural geometry governing optimal transport between Gaussian covariances is called the Bures-Wasserstein manifold (Bhatia et al., 2019; Takatsu, 2010). This Riemannian manifold defines the metric structure in the space of covariance matrices induced by the Wasserstein distance between Gaussian distributions, which characterizes optimality of transport between covariances. The geometry of this manifold can be described using the Lyapunov operator, which we define below. Definition 5.3 (Lyapunov Operator and Bures-Wasserstein Manifold). Given a covariance matrix Σ∈ S d ++ and tangent matrixU ∈T Σ S d ++ , the Lyapunov operatorL Σ [U] :T Σ S d ++ → S d ++ is the operator that returns the unique symmetric matrix A that solves: ΣA +AΣ = U(5.6) 119 Figure 9: Gaussian Schr ̈odinger Bridge Problem. The Gaussian SB between initial and terminal Gaussian distributionsπ 0 =N(μ,Σ) andπ T =N(μ ′ , Σ ′ ) reduces to solving an action minimization problem on the Bures– Wasserstein manifold of covariance matrices, where the covariance trajectory (Σ t ) t∈[0,T ] balances a kinetic energy term describing the rate of covariance change and a potential energy term induced by the diffusion of the reference process. which defines the Riemannian metric of the Bures-Wasserstein manifold given by: ⟨U,V⟩ Σ := 1 2 Tr(L Σ [U ]V ), U,V ∈T Σ S d ++ (5.7) whereU,V ∈T Σ S d ++ are two tangent vectors in the tangent space of the covariance matrices. Using the Lyapunov operator, the (Gaussian SB Problem) can be reformulated as an action minimization problem on the Bures-Wasserstein manifold of covariance matrices. This geometric formulation reveals that the optimal covariance trajectory evolves along curves minimizing a kinetic energy functional augmented by a diffusion correction term. Proposition 5.4 (Gaussian Schr ̈odinger Bridge Problem as Action Minimization (Theorem 2 of (Bunne et al., 2023))). The solution to the (Gaussian SB Problem) whereQis defined as pure Brownian motionQ:dX t =σ t dB t is equivalent to the solution of the action minimization problem on the Bures-Wasserstein manifold, defined as: inf (Σ t ) t∈[0,T ] Z T 0 1 2 ∥ ̇ Σ t ∥ 2 Σ t +U σ (Σ t ) dt s.t. ( Σ 0 = Σ Σ T = Σ ′ (5.8) whereU σ (Σ t ) := σ 4 t 8 Tr Σ −1 t is the potential energy that captures the entropic contribution of the diffusion term and∥ ̇ Σ t ∥ 2 Σ t =⟨ ̇ Σ t , ̇ Σ t ⟩ Σ is the Riemmannian metric defined in (5.7). Furthermore, the solution satisfies the Euler-Lagrange equation in Bures-Wasserstein geometry: ( ∇ ̇ Σ t ̇ Σ t =−grad − σ 4 t 8 TrΣ −1 t Σ 0 = Σ, Σ T = Σ ′ (Euler-Lagrange Equation) where∇ ̇ Σ t ̇ Σ t is the acceleration defined by the Riemannian gradient in the Bures-Wasserstein geometry. 120 Proof Sketch. While we don’t prove all required lemmas rigorously, we break down the key components of this proof in three steps 19 . Step 1: Define the Lyapunov Operator. A key component of this proof is that the Lyapunov operator L Σ t [ ̇ Σ t ] of the velocity of the covariance matrix along the curve Σ t takes the form: L Σ t [ ̇ Σ t ] = ̃ S t ⊤ Σ −1 t (5.9) where ̃ S t ∈ R d×d can be interpreted as the linear velocity field that transports the Gaussian distribution defined as: ̃ S t := tΣ ′ + (T − t)C σ − (T − t)Σ− tC ⊤ σ + σ t 2 (T − 2t)I d (5.10) Recall from Definition 5.3 that the Lyapunov operatorL Σ t [ ̇ Σ t ] is the symmetric matrixAthat solves Σ t A+AΣ t = ̇ Σ t and defines the Riemannian metric∥ ̇ Σ t ∥ 2 Σ t =Tr(AΣ t A). Therefore, kinetic energy term appearing in the action functional can be written as: 1 2 ∥ ̇ Σ t ∥ 2 Σ t = 1 2 Tr L Σ t [ ̇ Σ t ]Σ t L Σ t [ ̇ Σ t ] = 1 2 Tr ̃ S ⊤ t Σ −1 t Σ t |z I Σ −1 t ̃ S t = 1 2 Tr ̃ S ⊤ t Σ −1 t ̃ S t (5.11) Substituting this expression for the kinetic energy into the integrand of the action functional in (5.8) yields: 1 2 ∥ ̇ Σ t ∥ 2 Σ t +U σ (Σ t ) = 1 2 Tr ̃ S ⊤ t Σ −1 t ̃ S t + σ 4 t 8 Tr Σ −1 t = Tr 1 2 ̃ S ⊤ t Σ −1 t ̃ S t + σ 4 t 8 Σ −1 t (5.12) which yields an expression to the action functional written entirely in terms of the matrix ̃ S t and the inverse covariance Σ −1 t . Step 2: Use Continuity Equation to Derive Optimal Velocity. To finish the proof, we aim to show that the action minimization problem in (5.8) is equivalent to the (Gaussian SB Problem) with Q:=σ t B. To do this, we consider the special form of the (Gaussian SB Problem) with zero-mean marginals 20 : inf (p t ,v) Z T 0 E p t 1 2 ∥v(x,t)∥ 2 + σ 4 8 ∥∇ logp t (x)∥ 2 dt s.t. ( ∂ t p t =−∇· (p t v) p 0 =N (0, Σ), p T =N (0, Σ ′ ) (5.13) Then, the solution yields a Gaussian densityp t (x) =N(0,Σ t ) with zero-mean and covariance Σ t defined as: p t (x) = (2π) − 1 d (detΣ t ) − 1 2 exp − 1 2 x ⊤ Σ −1 t x (5.14) To derive the form of the optimal velocityv, we can expand both sides of the continuity constraint given by: ∂ t p t | z (⋆) =−∇· (p t v) |z (⋄) (5.15) For (⋆), we differentiate the Gaussian form ofp t (x) with respect to time using the Jacobi formula 21 , we get: ∂ t p t (x) = p t (x) 1 2 x ⊤ Σ −1 t ̇ Σ t Σ −1 t x− 1 2 Tr(Σ −1 t ̇ Σ t ) (⋆) 19 for rigorous proof, see (Bunne et al., 2023) Appendix B.2 20 since the mean follows the straight-line interpolation between the terminal marginals, we focus our attention on the covariance dynamics. 21 which states for any matrix valued functionA t :R ≥0 → R d×d , the time derivative of its determinant is equal to d dt det(A t ) = det(A t )· Tr(A −1 t )· d dt A t 121 For (⋄), we use the chain rule of divergence and the gradient of the log-density∇ logp t (x) =−Σ −1 t x to get: ∇· (p t v) =⟨∇p t ,v⟩ + p t (∇·v) =⟨p t ∇ logp t ,v⟩ + p t (∇·v) =⟨p t (−Σ −1 t x),v⟩ + p t (∇·v) = p t ⟨−Σ −1 t x,v⟩ +∇·v (⋄) Plugging (⋆) and (⋄) back into (5.15), we have: p t (x) 1 2 x ⊤ Σ −1 t ̇ Σ t Σ −1 t x− 1 2 Tr(Σ −1 t ̇ Σ t ) = p t (x) ⟨Σ −1 t x,v(x,t)⟩−∇·v(x,t) =⇒ ( ∇·v(x,t) = 1 2 Tr(Σ −1 t ̇ Σ t ) ⟨Σ −1 t x,v(x,t)⟩ = 1 2 ⟨Σ −1 t x, ̇ Σ t Σ −1 t x⟩ (5.16) which means that the optimal velocity field must be linear in x with the form: v(x,t) = ̃ S t Σ −1 t x(5.17) Step 3: Prove Equivalence of the Action Minimization and Gaussian SB Problem. Using the linear form of the velocity field in (5.17), the expectations appearing in the objective (5.13) can now be computed explicitly: 1 2 E[∥v(x,t)∥ 2 ] | z (⋆) + σ 4 8 E[∥∇ logp t (x)∥ 2 ] | z (⋄) (5.18) Sincex∼N(0,Σ t ), we haveE[x ⊤ ] = Σ t . Using this and the velocity field from (5.17), the kinetic energy term (⋆) can be written as: E[∥v(x,t)∥ 2 ] = E h Tr ̃ S ⊤ t Σ −1 t x ⊤ Σ −1 t ̃ S t i = Tr ̃ S ⊤ t Σ −1 t E h x ⊤ i |z Σ t Σ −1 t ̃ S t = Tr ̃ S ⊤ t Σ −1 t Σ t | z =I Σ −1 t ̃ S t = Tr ̃ S ⊤ t Σ −1 t ̃ S t (5.19) Similarly, using∇ logp t (x) =−Σ −1 t xandE[x ⊤ ] = Σ t =⇒ E[x ⊤ Ax] =Tr(AΣ t ), the score term (⋄) reduces to: E ∥∇ logp t (x)∥ 2 = E h ∥x ⊤ Σ −2 t x∥ 2 i = Tr(Σ −2 t Σ t ) = Tr(Σ −1 t )(5.20) Plugging (⋆) and (⋄) back into (5.18), we get: 1 2 Tr ̃ S ⊤ t Σ −1 t ̃ S t | z (⋆) + σ 4 8 Tr(Σ −1 t ) | z (⋄) = Tr 1 2 ̃ S ⊤ t Σ −1 t ̃ S t + σ 4 t 8 Σ −1 t (5.21) which exactly aligns with the form of the action function derived in (5.12), and we have shown that solving the action minimization problem is equivalent to solving the (Gaussian SB Problem). □ This result shows that the infinite-dimensional Gaussian SB Problem) reduces to a finite-dimensional action functional defined entirely on the trajectory of covariance matrices. Intuitively, we have shown that the optimal evolution of the covariance follows a curve on the manifold of symmetric positive definite matricesS d ++ where the first term measures the kinetic energy of the covariance transport in the Bures–Wasserstein geometry, and the second term acts as a potential induced by the diffusion term. The Schr ̈odinger bridge, therefore, corresponds to the curve that minimizes this action while matching the endpoint covariances Σ 0 = Σ and Σ T = Σ ′ . Next, we show that the solution for both problems yields a closed-form solution which can be written explicitly as an SDE. 122 Theorem 5.5 (Closed-Form Solution to Gaussian SB problem (Theorem 3 in Bunne et al. (2023))). The solution to the Gaussian SB problemP ⋆ with linear reference measureQdefined with the SDEdX Q t = (c t X t +α t )dt+σ t dB t is itself a Gaussian Markov stochastic processX 0:T where the intermediate marginals are Gaussiansp t =N(μ t ,Σ t ), with mean and covariance defined by: ( μ ⋆ t = ̄r t μ 0 + r t μ T +ζ(t)− r t ζ(T ) Σ ⋆ t = ̄r 2 t Σ 0 + r 2 t Σ T + r t ̄r t (C σ ⋆ +C ⊤ σ ⋆ ) + κ(t,t)(1− ρ t )I (5.22) s.t. r t := κ(t,T ) κ(T,T ) , r t := τ t − r t τ T , σ ⋆ := q τ −1 T κ(T,T ) ζ(t) := τ t R t 0 τ −1 s α s ds, ρ t := R t 0 τ −2 s σ 2 s ds R T 0 τ −2 s σ 2 s ds The time evolution of X t follows a closed-form SDE: dX t = f N (X t ,t)dt + σ t dB t ,s.t. f N (x,t) := S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t P t := ̇r t (r t Σ T + ̄r t C σ ⋆ ) Q t :=− ̇ ̄r t ( ̄r t Σ 0 + r t C σ ⋆ ) S t := P t −Q ⊤ t + (c t κ(t,t)(1− ρ t )− σ 2 t ρ t )I (5.23) where the matrix S ⊤ t Σ −1 t is symmetric. Proof. Following Bunne et al. (2023), we break down the proof in several steps. Step 1: Solving the Static Gaussian SB problem. First, we will solve the static Gaussian SB defined as the minimization problem: min P 0,T KL(P 0,T ∥Q 0,T ) : p 0 =N (μ 0 , Σ 0 ),p T =N (μ T , Σ T ) (5.24) whereQ 0,T is the endpoint law of the reference Gaussian process. By definition of linear SDEs of the formdX t = (c t X t +α t )dt+σ t dB t , the intermediate stateX t can be written as (Platen et al., 2010): X t = τ t X 0 + Z t 0 τ −1 s α s ds + Z t 0 τ −1 s σ t dB t , τ t := exp Z t 0 c s ds (5.25) Conditioned on an initial state X 0 ∼N (μ 0 , Σ 0 ), the mean and covariance of X t takes the form: η(t) := E[X t |X 0 ] = τ t X 0 + Z t 0 τ −1 s α s ds (5.26) κ(t,t ′ ) := E (X t −η(t))(X t −η(t ′ )) ⊤ X 0 = τ t τ t ′ Z t 0 τ −2 s σ 2 s ds I(5.27) Therefore, under the linear Gaussian measureQ, the probability of the pair (X 0 ,X T ) is a Gaussian defined by: Q(X T = x T |X 0 = x 0 ) =N (η(T ),κ(T,T )I) = (2π) − d 2 det(κ(T,T )I) − 1 2 exp − 1 2 (x T −η(T ))(κ(T,T )I) −1 (x T −η(T )) = (2π) − d 2 det(κ(T,T )I) − 1 2 exp − 1 2κ(T,T ) ∥x T − τ T x 0 −ζ(T )∥ 2 (5.28) 123 where the third equality follows from settingη(T) :=τ T x 0 +τ T R T 0 τ −1 s α s ds=τ T x 0 +ζ(T) and the fact that the covarianceκ(T,T)Iis isotropic. Now, we can expand the KL divergence and substitute (5.28) for Q 0,T to get: KL(P 0,T ∥Q 0,T ) = Z R d ×R d log dP 0,T dQ 0,T dP 0,T = Z R d ×R d (log dP 0,T )dP 0,T − Z R d ×R d (log dQ 0,T )dP 0,T = Z R d ×R d (log dP 0,T )dP 0,T − Z R d ×R d log (2π) − d 2 det(κ(T,T )I) − 1 2 exp − 1 2κ(T,T ) ∥x T − τ T x 0 −ζ(T )∥ 2 dP 0,T (x 0 ,x T ) = Z R d ×R d (log dP 0,T )dP 0,T − Z R d ×R d − d 2 log(2π)− 1 2 det(κ(T,T )I)− 1 2κ(T,T ) ∥x T − τ T x 0 −ζ(T )∥ 2 dP 0,T (x 0 ,x T ) = Z R d ×R d (log dP 0,T )dP 0,T + d 2 log(2π) + 1 2 det(κ(T,T )I) | z constant w.r.t. P 0,T + 1 2κ(T,T ) Z R d ×R d ∥x T − τ T x 0 −ζ(T )∥ 2 dP 0,T (x 0 ,x T ) = Z R d ×R d (log dP 0,T )dP 0,T + 1 2κ(T,T ) Z R d ×R d ∥x T − τ T x 0 −ζ(T )∥ 2 dP 0,T (x 0 ,x T ) + const.(5.29) Now, we aim to simplify this objective such that rather than matchingX T toτ T X 0 +ζ(T) overP 0,T , whereX 0 ,X T are sampled from two disjoint marginals, we can directly match the initial marginal transformed to timeTto the target marginal at timeT. Concretely, we define a transformed marginal ̃ P 0 defined by applying a change-of-variables: ̃ X 0 = τ T X 0 +ζ(T )∼N ( ̃ μ 0 , ̃ Σ 0 ) s.t. ( ̃ μ 0 = τ T μ 0 +ζ(T ) ̃ Σ 0 = τ 2 T Σ 0 (5.30) which yields the joint distribution ̃ P 0,T with marginals ̃ X 0 ∼ N( ̃ μ 0 , ̃ Σ 0 ) andX T ∼ N(μ T ,Σ T ). Since the change of variables is invertible and the mapP 0,T 7→ ̃ P 0,T is bijective, optimizing (5.29) over P 0,T is equivalent to optimizing over ̃ P 0,T . Under ̃ P 0,T , the quadratic cost becomes∥x T − τ T x 0 −ζ(T)∥ 2 =∥x T − ̃ x 0 ∥ 2 and the entropy term (log dP 0,T )dP 0,T is shifted by a constant that does not depend on P 0,T , yielding the KL divergence: KL( ̃ P 0,T ∥Q 0,T ) = Z R d ×R d (log d ̃ P 0,T )d ̃ P 0,T + 1 2κ(T,T ) Z R d ×R d ∥x T − ̃ x 0 ∥ 2 d ̃ P 0,T (x 0 ,x T ) + C (5.31) whereCis some constant. Since multiplying the objective by a constantκ(T,T)>0 does not change the minimizer, the optimization problem reduces to a standard entropic OT problem, defined as: min P 0,T KL(P 0,T ∥Q 0,T ) = min ̃ P 0,T KL( ̃ P 0,T ∥Q 0,T ) = min ̃ P 0,T Z R d ×R d 1 2 ∥x T − ̃ x 0 ∥ 2 d ̃ P 0,T + κ(T,T ) Z R d ×R d (log d ̃ P 0,T )d ̃ P 0,T (5.32) whereκ(T,T) is now the entropic regularization weight ̃σ 2 :=κ(T,T). From Lemma 5.1, the solution of (5.32) is given by the joint Gaussian: ̃ P ⋆ 0,T =N " μ 0 μ T # , " ̃ Σ 0 ̃ C ̃σ ̃ C ⊤ ̃σ ̃ Σ T #! ,s.t. ̃σ = p κ(T,T ) ̃ C ̃σ = 1 2 ̃ Σ 1 2 0 ̃ D ̃σ ̃ Σ − 1 2 0 − ̃σ 2 I ̃ D ̃σ = 4 ̃ Σ 1 2 0 Σ T ̃ Σ 1 2 0 − ̃σ 4 I 1 2 (5.33) 124 Applying the inverse transformX 0 =τ −1 T ( ̃ X 0 −ζ (T)) and the identityCov(X 0 ,X T ) =Cov(τ −1 T ̃ X 0 ,X T ) = τ −1 T Cov ( ̃ X 0 ,X T ), we recover the optimal joint GaussianP 0,T that solves the static Gaussian SB problem: π ⋆ 0,T =N " μ 0 μ T # , " Σ 0 τ −1 T ̃ C ̃σ τ −1 T ̃ C ⊤ ̃σ Σ T #! =N " μ 0 μ T # , " Σ 0 C σ ⋆ C ⊤ σ ⋆ Σ T #! (5.34) whereC σ ⋆ :=τ −1 T ̃ C ̃σ . From our definition ̃ Σ 0 =τ 2 T Σ 0 in (5.30) and ̃σ 2 :=κ(T,T), we expand ̃ D ̃σ as ̃ D ̃σ = 4(τ 2 T Σ 0 ) 1 2 Σ T (τ 2 T Σ 0 ) 1 2 − κ(T,T ) 2 I 1 2 = 4τ 2 T Σ 1 2 0 Σ T Σ 1 2 0 − κ(T,T ) 2 I 1 2 = τ T 4Σ 1 2 0 Σ T Σ 1 2 0 − κ(T,T ) 2 τ 2 T I | z D σ ⋆ 1 2 (5.35) which means σ 4 ⋆ := κ(T,T ) 2 τ 2 T and σ 2 ⋆ = κ(T,T ) τ T , proving the definition of σ ⋆ in Theorem 5.5. Step 2: Deriving the Endpoint-Conditioned Gaussian Bridge.In this step, we aim to derive a closed form solution for the distribution ofX t conditioned on the endpoint pair (X 0 ,X T ), where the stochastic process follows a linear SDE defined in (5.25). Conditioned on the initial state X 0 , we use the meanη(t) defined in (5.26) and covarianceκ(t,t ′ ) defined in (5.27) to write the joint Gaussian distribution of X t ,X T |X 0 as: X t ,X T |X 0 ∼N " μ 0 μ T # , " Σ 00 Σ 01 Σ 10 Σ 11 #! =N " η(0) η(T ) # , " κ(t,t)I d κ(t,T )I d κ(t,T )I d κ(T,T )I d #! (5.36) To compute the conditional distributionX t |X T from the joint Gaussian (X t ,X T ), we use the fact that any conditional distribution of a joint Gaussian random vector remains Gaussian, with mean and covariance given by: X t |X 0 ,X T ∼N ( ̃ μ, ̃ Σ),s.t. ( ̃ μ = μ 0 + Σ 01 Σ −1 11 (x−μ 1 ) ̃ Σ = Σ 00 − Σ 01 Σ −1 11 Σ 10 (5.37) Therefore, from (5.36), we have: E[X t |X 0 ,X T ] = η(t) + κ(t,T ) κ(T,T ) (X T −η(T )) (5.26) = τ t X 0 +ζ(t) + κ(t,T ) κ(T,T ) (X T − τ T X 0 −ζ(T )) = τ t − κ(t,T ) κ(T,T ) τ T |z =: ̄r t X 0 + κ(t,T ) κ(T,T ) | z =:r t X T +ζ(t)− κ(t,T ) κ(T,T ) | z =:r t ζ(T ) = ̄r t X 0 + r t X T +ζ(t)− r t ζ(T )(5.38) which means the expected bridge atX t is a weighted interpolation of the endpoints, with ̄r t defining the weight of the starting pointX 0 andr t defining the weight of the end point, with a deterministic correction term ζ(t)− r t ζ(T ). Similarly, we can derive the cross-covariance for any t ′ ≥ t as: E h (X t − E[X t |X 0 ,X T ]) (X t ′ − E[X t ′ |X 0 ,X T ]) ⊤ X 0 ,X T i = κ(t,t ′ )− κ(t,T )κ(t ′ ,T ) κ(T,T ) I d (5.39) 125 which follows from the covariance identityCov(X|Y) =Cov(X)− Cov(X,Y)Cov(Y) −1 Cov(Y ,X). Since the mean and covariance uniquely define a Gaussian process, the law of the conditional bridge is equivalent to the expected mean of the bridge with some random fluctuations defined by a zero-mean Gaussian random variable ξ t and covariance process (5.39): X t |X 0 ,X T law =E [X t |X 0 ,X T ] |z deterministic given X 0 ,X T +ξ t |z random fluctuation = ̄r t X 0 + r t X T +ζ(t)− r t ζ(T ) +ξ t (5.40) which proves that the Gaussian process conditioned on endpoints is simply a deterministic interpolation with independent residual noise ξ t . Step 3: Constructing the Gaussian SB from Endpoint Conditioned Bridges. Now, we can derive the solution to the Gaussian SB problem (5.2) as a mixture of endpoint conditioned briges, where the endpoints are sampled from the optimal joint distribution (X 0 ,X T )∼ P ⋆ 0,T for π ⋆ 0,T defined in (5.34) as the solution to the static Gaussian SB. First, we recall the (KL Divergence Chain Rule) which states: KL(P∥Q) = KL(P 0,T ∥Q 0,T ) + E P 0,T KL(P ·|0,T (·|X 0 ,X T )∥Q ·|0,T (·|X 0 ,X T )) (5.41) Therefore, the Gaussian SB can be constructed by first sampling (X 0 ,X T )∼ P ⋆ 0,T via (5.34) which yields the law: X t law = ̄r t X 0 + r t X T +ζ(t)− r t ζ(T ) +ξ t (5.42) where the covariance of the bridge measure is given by (5.27) with t ′ = t: Cov(ξ t ) (5.39) = κ(t,t)− κ(t,T ) 2 κ(T,T ) |z r t κ(t,T ) I d = κ(t,t) 1− κ(t,T ) 2 κ(T,T )κ(t,t) |z =:ρ t I =: κ(t,t)(1− ρ t )I(5.43) Applying the linearity of expectation, we derive the mean μ ⋆ t of the Gaussian SB as: μ ⋆ t = E P ⋆ 0,T [X t ] = ̄r t E P ⋆ 0,T [X 0 ] + r t E P ⋆ 0,T [X T ] +ζ(t)− r t ζ(T ) + E P ⋆ 0,T [ξ t ] = ̄r t μ 0 + r t μ T +ζ(t)− r t ζ(T )(5.44) Applying the covariance identity 22 , we derive the covariance Σ ⋆ t of the Gaussian SB as: Σ ⋆ t = Cov(X t ) = Cov ( ̄r t X 0 + r t X T +ζ(t)− r t ζ(T ) +ξ t ) = Cov ( ̄r t X 0 + r t X T ) + Cov (ξ t )(5.45) (5.43) = ̄r 2 t Σ 0 + r 2 t Σ T + r t ̄r t C σ ⋆ + r t ̄r t C ⊤ σ ⋆ +κ(t,t)(1− ρ t )I = ̄r 2 t Σ 0 + r 2 t Σ T + r t ̄r t (C σ ⋆ +C ⊤ σ ⋆ ) +κ(t,t)(1− ρ t )I(5.46) which concludes the first part of the proof that states the mean and covariance of the optimal marginal distributions of the Gaussian SB, given by: ( μ ⋆ t = ̄r t μ 0 + r t μ T +ζ(t)− r t ζ(T ) Σ ⋆ t = ̄r 2 t Σ 0 + r 2 t Σ T + r t ̄r t (C σ ⋆ +C ⊤ σ ⋆ ) + κ(t,t)(1− ρ t )I d (5.47) Since the endpoints are defined by a joint Gaussianπ ⋆ 0,T and the bridge is a linear interpolation of the endpoints with added Gaussian noiseξ t , we conclude that the optimal SB measure is indeed Gaussian. 22 Cov(aA + bB) = a 2 Cov(A) + b 2 Cov(B) + abCov(A,B) + abCov(B,A) 126 Furthermore, applying the ideas from Section 4.5, we have that the solution to the Gaussian SB problem is the unique measure P ⋆ that is both Markov and in the reciprocal class R(Q). Step 4: Deriving the SDE of the Gaussian Bridge. To simulate the solution to the Gaussian SB in practice, it is useful to derive the associated SDE which defines the forward propagation of the bridge given an intermediate state. Letφ:R d ×[0,T]→ Rbe a smooth test function on the SB processX 0:T . Then, the core of this derivation is to show thatφ(X t ) is an Itˆo process satisfying Itˆo’s formula (2.7) with a generatorA t of the form: A t φ(x,t) := ∂ t φ(x,t) +f N (x,t) ⊤ ∇φ(x,t) + σ 2 t 2 ∆φ(x,t)(5.48) for a control driftf N :R d ×[0,T]→ Rthat defines the SDE ofX t given bydX t =f N (X t ,t)dt+σ t dB t . To do this, we must match the time derivative ofφ(X t ,t) with (5.48) by taking the continuous time limit: lim ∆t→0 E[φ(X t+∆t ,t + ∆t)|X t = x]− φ(x,t) ∆t (5.49) The first step is to deriveE[φ(X t+∆t ,t+ ∆t)|X t =x] and take the infinitesimal time limit ∆t→0. Since the conditional expectation is a Gaussian, we aim to derive its meanμ t+∆t :=E[X t+∆t |X t = x] =μ t + Σ t,t+∆t Σ −1 t (x − μ t ) (where Σ t,t+∆t denotes the cross-covariance) and covariance Σ t+∆t :=Cov(X t+∆t |X t =x) using the first-order approximation. First, we compute the time derivative ̇ Σ ⋆ t as: ̇ Σ ⋆ t := ∂ t Σ ⋆ t = ∂ t ̄r 2 t Σ 0 + r 2 t Σ T + r t ̄r t (C σ ⋆ +C ⊤ σ ⋆ ) + κ(t,t)− κ(t,T ) 2 κ(T,T ) I = 2 ̄r t ̇ ̄r t Σ 0 + 2r t ̇r t Σ T + ( ̇r t ̄r t + r t ̇ ̄r t )(C σ ⋆ +C ⊤ σ ⋆ ) + ∂ t κ(t,t)− 2κ(t,T )∂ t κ(t,T ) κ(T,T ) |z = ̇r t I = ̇r t (r t Σ T + ̄r t C σ ⋆ ) | z P t + ̇r t r t Σ T + ̄r t C ⊤ σ ⋆ | z P ⊤ t + ̇ ̄r t ( ̄r t Σ 0 + r t C σ ⋆ ) | z −Q t + ̇ ̄r t ̄r t Σ 0 + r t C ⊤ σ ⋆ | z −Q ⊤ t + ∂ t κ(t,t)− 2 ̇r t κ(t,T ) I = P t +P ⊤ t − Q t +Q ⊤ t + ∂ t κ(t,t)− 2 ̇r t κ(t,T ) I(5.50) Next, we derive the cross-covariance Σ t,t+∆t using the definition in (5.39) given by Σ t,t+∆t := E (X t −μ t )(X t+∆t −μ t+∆t ) ⊤ where we make the following decomposition: X t −μ t (5.42) = ̄r t (X 0 −μ 0 ) + r t (X T −μ T ) +ξ t (5.51) X t+∆t −μ t+∆t (5.42) = ̄r t+∆t (X 0 −μ 0 ) + r t+∆t (X T −μ T ) +ξ t+∆t (5.52) 127 Setting A := X 0 −μ 0 , B := X T −μ T , we can write the cross covariance as: Σ t,t+∆t = E h ( ̄r t A + r t B +ξ t )( ̄r t+∆t A + r t+∆t B +ξ t+∆t ) ⊤ i = ̄r t ̄r t+∆t E[A ⊤ ] | z =:Σ 0 + ̄r t r t+∆t E[AB ⊤ ] |z =:C σ ⋆ + ̄r t E[Aξ ⊤ t+∆t ] | z =0 +r t ̄r t+∆t E[BA ⊤ ] |z =:C ⊤ σ ⋆ + r t r t+∆t E[B ⊤ ] |z Σ T +r t E[Bξ ⊤ t ] |z =0 + ̄r t+∆t E[ξ t A ⊤ ] | z =0 +r t+∆t E[ξ t B ⊤ ] | z =0 +E[ξ t ξ ⊤ t+∆t ] (5.39) = ̄r t ̄r t+∆t Σ 0 + ̄r t r t+∆t C σ ⋆ + r t ̄r t+∆t C ⊤ σ ⋆ + r t r t+∆t Σ T + κ(t,t + ∆t)− κ(t,T )κ(t + ∆t,T ) κ(T,T ) I = ̄r t ̄r t+∆t Σ 0 + ̄r t r t+∆t C σ ⋆ + r t ̄r t+∆t C ⊤ σ ⋆ + r t r t+∆t Σ T + (κ(t,t + ∆t)− r t+∆t κ(t,T ))I(5.53) where all covariances with only a single noise termξ t vanish since the noise is independent ofX 0 ,X T . Since Σ t has the same structure as Σ t+∆t , we can write the cross covariance as Σ t,t+∆t = Σ t + ∆Σ, where ∆Σ is the change in covariance from t→ t + ∆t, which gives the following expression: Σ t,t+∆t =Σ t + ̄r t ( ̄r t+∆t − ̄r t )Σ 0 + ̄r t (r t+∆t −r t )C σ ⋆ + r t ( ̄r t+∆t − ̄r t )C ⊤ σ ⋆ + r t (r t+∆t −r t )Σ T + (κ(t,t + ∆t)−κ(t,t)− r t+∆t κ(t,T )+r t κ(t,T ))I =Σ t + ( ̄r t+∆t − ̄r t ) ̄r t Σ 0 + r t C ⊤ σ ⋆ + (r t+∆t − r t ) ( ̄r t C σ ⋆ + r t Σ T ) + (κ(t,t + ∆t)−κ(t,t)− r t+∆t κ(t,T ) + r t κ(t,T ))I =Σ t + ̄r t+∆t − ̄r t ̇r t P t − r t+∆t − r t ̇ ̄r t Q ⊤ t + (κ(t,t + ∆t)−κ(t,t)− r t+∆t κ(t,T )+r t κ(t,T ))I (5.54) Applying the first-order approximation which preserves all terms scaled byhand denoting the higher-order terms with o(∆t), we have: Σ t,t+∆t = Σ t + ∆t P t −Q ⊤ t + ∂κ dt ′ (t,t)− ̇r t κ(t,T ) I |z =:S t +o(∆t)(5.55) = Σ t + ∆t P t −Q ⊤ t + ∂ t τ t τ t ′ Z t 0 τ −2 s σ 2 s ds t ′ =t − 1 κ(T,T ) ∂ t τ t τ T Z t 0 τ −2 s σ 2 s ds κ(t,T ) I + o(∆t) = Σ t + ∆t P t −Q ⊤ t + τ t ̇τ t Z t 0 τ −2 s σ 2 s ds |z c t κ(t,t) − 1 κ(T,T ) ̇τ t τ T Z t 0 τ −2 s σ 2 s ds |z c t κ(t,T ) +τ t τ T ∂ t Z t 0 τ −2 s σ 2 s ds | z =τ −2 t σ 2 t (Leibniz rule) κ(t,T ) I + o(∆t) = Σ t + ∆t P t −Q ⊤ t + c t κ(t,t)− c t κ(t,T ) κ(T,T ) + τ T σ 2 t τ t κ(T,T ) κ(t,T ) I + o(∆t) = Σ t + ∆t P t −Q ⊤ t + c t κ(t,t)− c t κ(t,T ) 2 κ(T,T ) |z c t ρ t κ(t,t) +σ 2 t τ T κ(t,T ) τ t κ(T,T ) |z =:ρ t I + o(∆t) = Σ t + ∆t P t −Q ⊤ t + c t κ(t,t)(1− ρ t ) + σ 2 t ρ t I |z =:S t + o(∆t) = Σ t + ∆tS t + o(∆t)(5.56) which gives us the expression for the cross covariance Σ t,t+∆t = Σ t + ∆tS t +o(∆t). Now, we can expand the mean ̃ μ t+∆t and covariance ̃ Σ t+∆t of the conditional processX t+∆t |X t using the same logic as (5.37) to get: ̃ μ t+∆t = μ t+∆t + Σ ⊤ t,t+∆t Σ −1 t (x−μ t )(5.57) ̃ Σ t+∆t = Σ t+∆t + Σ ⊤ t,t+∆t Σ −1 t Σ t,t+∆t (5.58) 128 Starting with the mean ̃ μ t+∆t , we substitute (5.56) to get: ̃ μ t+∆t = μ t+∆t + (Σ t + ∆tS t ) ⊤ Σ −1 t (x−μ t )+o(∆t) = μ t + ∆t ̇ μ t + (Σ ⊤ t Σ −1 t | z =I +hS ⊤ t Σ −1 t )(x−μ t )+o(∆t) = μ t + ∆t ̇ μ t +x−μ t + (hS ⊤ t Σ −1 t )(x−μ t ) + o(∆t) = x + h(S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t ) + o(∆t) (5.59) Then, substituting (5.56) into the covariance, we get: ̃ Σ t+∆t = Σ t+∆t − (Σ t + ∆tS t ) ⊤ Σ −1 t (Σ t + ∆tS t ) + o(∆t) = Σ t + ∆t ̇ Σ t − (Σ t + ∆tS t ) ⊤ Σ −1 t (Σ t + ∆tS t ) + o(∆t) = Σ t + ∆t ̇ Σ t − (I + hS ⊤ t Σ −1 t )(Σ t + ∆tS t ) + o(∆t) = Σ t + ∆t ̇ Σ t − (Σ t + ∆tS t + ∆tS ⊤ t + h 2 S ⊤ t Σ −1 t S t | z o(∆t) ) + o(∆t) = h( ̇ Σ t −S t −S ⊤ t ) + o(∆t)(5.60) and substituting the definition for S t from (5.55) and ̇ Σ t from (5.50), we get: ̃ Σ t+∆t (5.55,5.50) = h P t +P ⊤ t − Q t +Q ⊤ t + ∂ t κ(t,t)− 2 ̇r t κ(t,T ) I −P t −Q ⊤ t + ∂κ dt ′ (t,t)− ̇r t κ(t,T ) I − P t −Q ⊤ t + ∂κ dt ′ (t,t)− ̇r t κ(t,T ) I ⊤ + o(∆t) = h ∂ t κ(t,t)− 2 ∂κ dt ′ (t,t) I + o(∆t)(5.61) Now, substituting (5.27) and decomposing ∂κ dt ′ (t,t)− ̇r t κ(t,T) using the same procedure as (5.55-5.56), we have: ̃ Σ t+∆t = h ∂ t τ 2 t Z t 0 τ −2 s σ 2 s ds − 2τ t ̇τ Z t 0 τ −2 s σ 2 s ds I + o(∆t) = h 2 ̇τ t τ t Z t 0 τ −2 s σ 2 t ds + σ 2 t − 2τ t ̇τ Z t 0 τ −2 s σ 2 s ds I + o(∆t) = hσ 2 t I + o(∆t) (5.62) Finally, we are ready to derive the form ofE[φ(X t+∆t ,t+ ∆t)|X t =x] as the expected value at X t+∆t . Since X t+∆t |X t = x is Gaussian with mean ̃ μ t+∆t and covariance ̃ Σ t+∆t , we can write: X t+∆t = ̃ μ t+∆t + ̃ Σ 1/2 t+∆t z, z ∼N (0,I d )(5.63) Applying the second-order Taylor expansion of φ( ̃ x + ̃ μ t+∆t ,t + ∆t) around ̃ x = 0 yields: φ( ̃ x + ̃ μ t+∆t ,t + ∆t) = φ( ̃ μ t+∆t ,t + ∆t) +∇φ ⊤ ̃ x + 1 2 ̃ x ⊤ ∇ 2 φ ̃ x + o(∥ ̃ x∥ 2 )(5.64) Given the Gaussian identitiesE[z] = 0 andE[z ⊤ ] =I d , it follows thatE[ ̃ x] = 0 andE[ ̃ x ̃ x ⊤ ] = ̃ Σ t+∆t . Using these identities and taking the conditional expectation, we have: E[φ(X t+∆t ,t + ∆t)|X t = x] = φ( ̃ μ t+∆t ,t + ∆t) + 1 2 Tr ̃ Σ t+∆t ∇ 2 φ( ̃ μ t+∆t ,t + ∆t) + o(∆t) (5.62) = φ( ̃ μ t+∆t ,t + ∆t) + hσ 2 t 2 ∆φ( ̃ μ t+∆t ,t + ∆t) + o(∆t)(5.65) 129 where the second line follows from ̃ Σ t+∆t =hσ 2 t I+o(∆t) from (5.62). From (5.59), the mean expands to ̃ μ t+∆t =x+h(S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t ) +o(∆t) =x+O(∆t), wherehis some infinitesimal time increment, we can apply the Taylor expansion around x to get: φ( ̃ μ t+∆t ,t + ∆t) = φ(x,t + ∆t) + h ∇φ(x,t + ∆t),S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t + o(∆t)(5.66) Then, substituting the expression (5.66) into (5.65), we have: E[φ(X t+∆t ,t + ∆t)|X t = x] = φ( ̃ μ t+∆t ,t + ∆t) + hσ 2 t 2 ∆φ( ̃ μ t+∆t ,t + ∆t) + o(∆t) = φ(x,t + ∆t) + h ∇φ(x,t + ∆t),S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t + hσ 2 t 2 ∆φ(x,t + ∆t) + o(∆t) = φ(x,t + ∆t) + h σ 2 t 2 ∆φ(x,t + ∆t) + ∇φ(x,t + ∆t),S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t + o(∆t) (5.67) Dividing both sides byhand taking the limit ash→0, we recover the final form of the generator A t for φ(x,t) as: A t φ(x,t) = lim ∆t→0 E[φ(X t+∆t ,t + ∆t)|X t = x]− φ(x,t) h = lim ∆t→0 φ(x,t + ∆t)− φ(x,t) h + σ 2 t 2 ∆φ(x,t + ∆t) + ∇φ(x,t + ∆t),S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t +O(∆t) = ∂ t φ(x,t) + σ 2 t 2 ∆φ(x,t + ∆t) + ∇φ(x,t + ∆t),S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t (5.68) where the control drift takes the form: f N (x,t) := S ⊤ t Σ −1 t (x−μ t ) | z shape-correcting term + ̇ μ t |z drift of mean (Gaussian SB Drift) which can be decomposed into the mean drift of the Gaussian and a shape-correcting term that consists of the deviation from the mean (x−μ t ), how expensive it is under the covariance Σ −1 t , and how strongly to correct it S ⊤ t . As we show in Proposition 2.23, the optimal control drift of the non-linear SB problem can be written as a gradient field asf N (x,t) =∇ψ t (x) of the Lagrange multiplierψ t (x) such that the Jacobian of f N is symmetric or ∂ i f j = ∂ j f i . Since we have: ∇f N (x,t) =∇(S ⊤ t Σ −1 t (x−μ t ) + ̇ μ t ) = S ⊤ t Σ −1 t (5.69) this implies that S ⊤ t Σ −1 t is symmetric and concludes our proof.□ From this result, we establish that, when the reference dynamics are linear–Gaussian, the (Gaussian SB Problem) admits an exact closed-form solution that remains within the class of Gaussian Markov processes, which is fully characterized by finite-dimensional evolutions of the mean and covariance and by an affine drift field that admits a gradient-field structure. This reveals the core insight that entropy-regularized transport between Gaussian marginals preserves the Gaussian structure of the distribution and induces a potential-driven flow that conserves distributional structure. This Gaussian case serves as a solvable model that provides a foundation for the upcoming sections on more specialized variations of the SB problem. 130 Figure 10: Generalized Schr ̈odinger Bridge Problem. The generalized SB problem extends the dynamic SB problem by introducing a mean-field interaction costI(x,p t ) that depends on the evolving marginal distribution. The resulting objective balances kinetic energy 1 2 ∥u(x,t)∥ 2 with an interaction energy term integrated against the state densityp t (x). The optimal dynamics transport particles between the prescribed marginals (π 0 ,π T ) while accounting for collective interactions encoded byI(x,p t ), leading to a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system that characterizes the optimal bridge. 5.2 Generalized Schr ̈odinger Bridge Problem Prerequisite: Section 2.7, 2.8, 4.3 Up to this point, we have considered particles as acting independently via an optimal path along the Schr ̈odinger bridge such that the total distribution over many particles matches the endpoint marginals. However, in many settings, particles evolve via stochastic trajectories that depend not only on their individual state and control but also interactions with the population distribution p t . Solving for the optimal evolution of particles is referred to as solving a Mean-Field Game, since the particles are influenced by the average dynamics of the population. At equilibrium, each particle evolves via optimal control given the density of the particlesp t , and the populationp t is generated from these optimally controlled particles. To determine the optimal dynamics, we leverage the definition of the value functionψ t (x) :R d ×[0,T]→ Rwhich defines the minimum cost of transportingxat timet. The gradient of the function∇ψ t (x)∈ R d defines a potential field that adjusts the drift of the reference process so that the evolving distribution satisfies the endpoint constraints. The value functionψ t and marginal densityp t are coupled via a Hamiltonian functionH(x,∇ψ t ,p t ) : R d × R×P(R d )→ R, which describes the dynamics of the interacting particles and an interaction functionI(x,p t ) :R d ×P(R d )→ Rthat can be defined depending on the task. Given these functions, the pair of optimal value function and optimal state PDF (ψ t ,p ⋆ t ) solves the following pair of PDEs: ( ∂ t ψ t +H(x,∇ψ t ,p ⋆ t ) + σ 2 t 2 ∆ψ t =I(x,p ⋆ t )ψ T (x) = Φ(x) ∂ t p ⋆ t +∇· (p ⋆ t ∇ ∇ψ t H(x,∇ψ t ,p ⋆ t ))− σ 2 t 2 ∆p ⋆ t = 0 p 0 = π 0 ,p T = π T (5.70) where Φ(x) :R d → Ris a terminal cost that can be defined to enforce a target distribution in the case of Schr ̈odinger bridges. Given a solution pair (ψ t ,p ⋆ t ), the dynamics of each particle evolve via the SDE defined by: dX t =−∇ ∇ψ t (X t ) H(X t ,∇ψ t (X t ),p t )dt + σ t dB t , X 0 ∼ π 0 (5.71) which satisfies the marginals defined by p t as the number of particles goes to infinity. Now that we have defined the Mean-Field Game problem, we can observe that the coupled PDEs in (5.70) resemble the (HJB-FP System) defined for the dynamic SB problem in Proposition 2.23 with 131 the additional mean field interactionI(x,p t ) :R d ×P(R d )→ R. Therefore, we can leverage this connection between mean-field games and dynamic SB to define the Generalized Schr ̈odinger Bridge Problem (Chen, 2023; Chen et al., 2015; Liu et al., 2022; 2023a). Definition 5.6 (Generalized Schr ̈odinger Bridge Problem). Given an interaction costI(x,p t ) : R d × P(R d )→ R, reference driftf(x,t), diffusion coefficientσ t , and terminal marginal constraints π 0 ,π T ∈P(R d ), the generalized SB problem can be written as: inf u E X 0,T ∼P u Z T 0 1 2 ∥u(X t ,t)∥ 2 +I(X t ,p t ,t) dt (Generalized SB Problem) s.t. ( dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB t X 0 ∼ π 0 , X T ∼ π T (5.72) which can also be written as the density-space objective: inf (u,p t ) Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 +I(x,p t ,t) p t (x)dxdt (5.73) s.t. ( ∂ t p t (x) =−∇· p t (x)(f (x,t) + σ t u(x,t)) + σ 2 t 2 ∆p t p 0 = π 0 , p T = π T (5.74) Following the derivation of the nonlinear HJB-FP system in Section 2.7, which defines the optimal pair (ψ t ,p ⋆ t ) and the Hopf-Cole transform in Section 2.8, which defines the optimal potentials (φ t , ˆφ t ), we can derive the optimality conditions of the generalized SB problem. Definition 5.7 (Optimality Conditions of Generalized Schr ̈odinger Bridge Problem). Defining the Hamiltonian as: H(x,∇ψ t ,p t ) := 1 2 ∥σ t ∇ψ t ∥ 2 −∇ψ ⊤ t f (x,p t ,t)(5.75) We can write the optimality conditions (ψ t ,p ⋆ t ) as the pair of coupled non-linear HJB-FP system with the interaction term I(x,p ⋆ t ) : R d ×P(R d )→ R given by: ( ∂ t ψ t + σ 2 t 2 ∥∇ψ t ∥ 2 +⟨∇ψ t ,f⟩ + σ 2 t 2 ∆ψ t =I(x,p ⋆ t ) ∂ t p ⋆ t +∇· (p ⋆ t (f + σ 2 t ∇ψ t ))− σ 2 t 2 ∆p ⋆ t = 0 s.t. ( p 0 = π 0 p T = π T (5.76) To transform the system of non-linear PDEs to linear PDEs, we can apply the Hopf-Cole transform discussed in Section 2.8, which defines the following change-of-variables: ψ t (x) = logφ t (x), p ⋆ t (x) = φ t (x) ˆφ t (x)(5.77) which satisfy the pair of linear PDEs: ( ∂ t φ t =−⟨∇φ t ,f⟩− σ 2 t 2 ∆φ t +Iφ t ∂ t ˆφ t =−∇· ( ˆφ t f ) + σ 2 t 2 ∆ ˆφ t −I ˆφ t s.t. ( π 0 = φ 0 ˆφ 0 π T = φ T ˆφ T (5.78) which differs from the linear PDEs from (Hopf-Cole PDEs) only by the interaction term I(x,p ⋆ t ). 132 In Section 4.3, we established a pair of forward-backward SDEs that define the evolution of the dynamic Schr ̈odinger bridge problem, which describes the evolution of the Schr ̈odinger potentials (φ t , ˆφ t ) from the system of linear PDEs derived via the Hopf-Cole transform (Section 2.8). Given the modified Hopf-Cole linear PDEs for the generalized SB problem, we must define a new set of FBSDEs that integrates the mean-field interaction term I. Proposition 5.8 (Generalized SB Forward-Backward SDEs (Theorem 2 in Liu et al. (2022))). Given the Schr ̈odinger potentials (φ t , ˆφ t ) that solve the Hopf-Cole linear PDEs from (5.78) and a stochastic processX 0:T that satisfies the forward-time SDE, we define additional stochastic processes as: Y t = Y (X t ,t) := logφ t (X t ),Z t = Z(X t ,t) := σ t ∇ logφ t (X t )(5.79) b Y t = b Y (X t ,t) := log ˆφ t (X t ), b Z t = b Z(X t ,t) := σ t ∇ log ˆφ(X t )(5.80) Then, the forward time evolutiont∈[0,T] ofX 0:T , (Y t ) t∈[0,T ] and ( b Y t ) t∈[0,T ] are characterized by the FBSDEs: dX t = (f (X t ,p ⋆ t ) + σ t Z t )dt + σ t dB t dY t = 1 2 ∥Z t ∥ 2 +I(X t ,p ⋆ t ) dt +Z t ⊤ dB t d b Y t = 1 2 ∥ b Z t ∥ 2 +∇· (σ t b Z t −f (X t ,p ⋆ t ))− b Z t ⊤ Z t −I(X t ,p ⋆ t ) dt + b Z t ⊤ dB t (5.81) To condition the dynamics on the target distributionπ T , on the time-reversed coordinate s:=T−t∈[0,T], we define the time-reversed SDE ( ̃ X s ) s∈[0,T ] and the corresponding FBSDEs for (Y s ) s∈[0,T ] and ( b Y s ) s∈[0,T ] as: d ̃ X s = (−f ( ̃ X s ,p ⋆ s ) + σ s b Z s )ds + σ t dB s dY s = 1 2 ∥Z s ∥ 2 +∇· (σ t Z s +f ( ̃ X s ,p ⋆ s ))−Z s ⊤ b Z s −I( ̃ X s ,p ⋆ s ) dt +Z s ⊤ dB s d b Y s = 1 2 ∥ b Z s ∥ 2 +I(X s ,p ⋆ s ) ds + b Z s ⊤ dB s (5.82) Given the SB optimality conditionp ⋆ t =φ t ˆφ t , we define the interaction termI:R d ×P(R d )→ R as: I(X t ,p ⋆ t ) =I(X t ,φ t ˆφ t ), f (X t ,p ⋆ t ) = f (X t ,φ t ˆφ t )(5.83) I( ̃ X s ,p ⋆ s ) =I( ̃ X s ,φ s ˆφ s ), f ( ̃ X s ,p ⋆ s ) = f ( ̃ X s ,φ s ˆφ s )(5.84) Proof Sketch. This proof follows closely from the derivation in Section 4.3, where we apply Itˆo’s formula to getd logφ t andd log ˆφ t , but instead of substituting the Hopf-Cole transform for the standard dynamic SB problem for∂ t φ t and∂ t ˆφ t , we substitute the Hopf-Cole transform with the interaction termIgiven in (5.78). The reverse-time FBSDEs follow the same steps after reformulating the HJB-FP PDEs with the reversed time coordinates:=T−tand applying the Hopf-Cole transform. □ Compared to the dynamic SB FBSDEs derived in Section 4.3, the only structural modification appears through the interaction termI(x,p ⋆ t ), which introduces a mean-field dependence into both the forward and backward SDEs. The forward–backward structure remains similar to the dynamic SB FB-SDEs, but the system now encodes collective effects through the optimal marginal flow defined as p ⋆ t = φ t ˆφ t . 133 Figure 11: Multi-Marginal Schr ̈odinger Bridge Problem. The multi-marginal Schr ̈odinger bridge constructs an optimal stochastic process whose marginal distributions match a sequence of prescribed intermediate densities π t k K k=0 . Starting from a reference stochastic processQ, the optimal path measureP ⋆ is obtained by minimizing the path-space KL divergence while matching the marginal constraints. Remark 5.9 (Generalization of Dynamic SB). When the interaction term vanishes for all (x,t)∈ R d ×[0,T], i.e.,I ≡0, and the reference drift is independent of the marginal, i.e., f(x,p t ,t) =f(x,t), the generalized SB problem reduces to (Dynamic SB Problem) and can be interpreted as a generalization of dynamic SB to McKean-Vlasov settings with mean-field interactions. In summary, the generalized Schr ̈odinger bridge extends the classical dynamic SB formulation by introducing an interaction termI(x,p t ), that allows the dynamics to depend on the evolving marginal distributionp t . Crucially, we show that the extension of the dynamic SB problem to settings with mean-field interactions preserves the fundamental structure of the dynamic SB problem with the only difference being an additive interaction term. Therefore, many frameworks used to solve the dynamic SB problem can be adapted to solve the generalized SB problem. 5.3 Multi-Marginal Schr ̈odinger Bridge Problem The standard dynamic SB problem aims to determine the optimal bridge that maps particles from the initial distributionπ 0 to the terminal distributionπ T while minimizing the KL divergence to the reference measure. A natural extension of this problem is to consider multiple marginal constraints at multiple points along the time horizon, which can be applied to construct feasible trajectories between observed snapshots over coarse time intervals. This variation of the SB problem is considered the Multi-Marginal Schr ̈odinger Bridge Problem (Chen et al., 2023; Chen et al., 2019; Theodoropoulos et al., 2025). Definition 5.10 (Standard Multi-Marginal Schr ̈odinger Bridge Problem). Given a uncontrolled reference measureQmultiple marginal constraintsπ t k ∈P(R d ) K k=1 at sequential time points 0 =t 0 < · < t k < · < t K =T, the standard multi-marginal Schr ̈odinger bridge 134 problem aims to determine a control u(x,t) that minimizes: inf u E X 0:T ∼P u Z T 0 1 2 ∥u(X t ,t)∥ 2 dt (Multi-Marginal SB Problem) s.t. ( dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB t X t k ∼ π t k ,∀k ∈0,...,K (5.85) where the optimalu ⋆ generates the path measureP ⋆ of minimal relative entropy with respect Qamong all controlled measures matching the prescribed marginals. This can equivalently be written as the density-space objective: inf u E X 0:T ∼P u " Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 p t (x)dxdt # (5.86) s.t. ( ∂ t p t (x) =−∇· p t (x)(f (x,t) + σ t u(x,t)) + σ 2 t 2 ∆p t p t k = π t k ,∀k ∈0,...,K (5.87) Since intermediate marginal state distributions can have associated velocity information that deter- mines how it propagates to the subsequent distribution such thatπ t k (x,v)∈P(R 2d ), we re-formulate the multi-marginal SB problem in phase space, known as the Momentum Multi-Marginal Schr ̈odinger Bridge problem (Chen et al., 2023; Theodoropoulos et al., 2025), where the state is given by a vector (x,v)∈ R 2d . In this setting, the reference dynamics that follow a pair of second-order SDEs: ( dx t = v t dt dv t = a t dt + σ t dB t (5.88) where stochasticity is introduced into the second-order SDE. We define the marginal distribution generated by the SDEs in (5.88) over the phase space asp t ∈P(R 2d ), the distribution of the state as μ(x,t) = R R d p t (x,v)dv, and the distribution over velocity as ξ(v,t) = R R d p t (x,v)dx. Definition 5.11 (Momentum Multi-Marginal Schr ̈odinger Bridge Problem). Given multiple marginal distributions over the phase spaceπ t 0 ,...,π t K ∈P (R 2d ) defined at sequential time pointst 0 ,...,t K ∈[0,T] on the time horizon, the multi-marginal SB problem aims to determine the optimal acceleration a ⋆ t ≡ a ⋆ t (x,v,t)∈ R 2d that solves the SOC problem: a ⋆ t = arg min a t E p t Z T 0 1 2 ∥a t ∥ 2 dt s.t. dx t = v t dt dv t = a t dt + σ t dB t (x t k ,v t k )∼ π t k ,∀k ∈0,...,K (5.89) Equivalently, writing the problem in terms of the distributionp t and the Fokker-Planck equation from Section 2.4, we have: a ⋆ t = arg min a t E p t " Z T 0 Z R d Z R d 1 2 ∥a t ∥ 2 p t dxdvdt # (5.90) s.t. ( ∂ t p t (x,v) +v·∇p t +∇ v · (a t p t )− σ 2 t 2 ∆ v p t = 0 μ(x,t k ) = R R d π t k (x,v)dv,∀k ∈0,...,K (5.91) The diffusion term in the FP equation makes the dynamics irreversible since diffusion increases entropy over a single time direction. To absorb the diffusion term into a deterministic drift, we can 135 expand the velocity term σ 2 t 2 ∆ v p t as: ∆ v p t =∇ v · (∇ v p t ) =∇ v · (p t ∇ v logp t )(5.92) Substituting this into the FP equation in (5.91), we get: ∂ t p t (x,v) +v·∇p t +∇ v · a t p t − σ 2 t 2 p t ∇ v logp t = 0 ∂ t p t (x,v) +v·∇p t +∇ v · a t − σ 2 t 2 ∇ v logp t | z =: ˆ a t p t = 0(5.93) Then, we can define ˆ a t := a t − σ 2 t 2 ∇ v logp t to get: ∂ t p t (x,v) +v·∇p t +∇ v · ( ˆ a t p t ) = 0(5.94) To write the cost functional in terms of ˆ a, we substitutea t = ˆ a t + σ 2 t 2 ∇ v logp t in the squared cost to get: 1 2 ∥a∥ 2 = 1 2 ˆ a + σ 2 t 2 ∇ v logp t 2 = 1 2 ∥ ˆ a∥ 2 + σ 2 t 8 ∥∇ v logp t ∥ 2 + 1 2 ⟨ ˆ a,∇ v logp t ⟩(5.95) and integrating the expanded form over time and phase space, we have: Z T 0 Z 1 2 ∥ ˆ a∥ 2 p t dxdvdt | z transport cost + Z T 0 Z σ 2 t 8 ∥∇ v logp t ∥ 2 p t dxdvdt | z velocity Fisher information + Z T 0 Z 1 2 ⟨ ˆ a,∇ v logp t ⟩p t dxdvdt | z cross term (5.96) where the first term is the transport cost. The second term is the velocity Fisher information (squared norm of the score function), which measures the sensitivity of the distribution under infinitesimal changes to the velocity. Since the diffusion only acts on the velocity coordinate in (5.88), this term acts as an uncertainty regularization after absorbing the diffusion into the deterministic drift. Finally, the cross term can be expanded as follows: Z T 0 Z 1 2 ⟨ ˆ a,∇ v logp t ⟩p t dxdvdt = Z T 0 Z 1 2 ˆ a· (p t ∇ v logp t )dxdvdt = Z T 0 Z 1 2 ˆ a·∇ v p t dxdvdt(5.97) We now observe that this cross term is the alignment of the transport velocity ˆ awith the density gradient∇ v p t in velocity space. Using the identity∇ v ·(p t ˆ a) = ˆ a·∇ v p t +p t (∇ v · ˆ a) and applying integration by parts in velocity space under vanishing boundary conditions 23 , we get: Z T 0 Z R d 1 2 ( ˆ a·∇ v p t )dvdxdt = Z T 0 Z 1 2 (∇ v · (p t ˆ a))dvdxdt− Z T 0 Z 1 2 p t (∇ v · ˆ a)dvdxdt | z =0 (divergence integrates to 0) (5.98) where the divergence∇ v · ˆ aintegrated overR d is 0 since probability mass is conserved, and since there are no boundaries on velocity, the flux must be 0. Now, substituting the continuity equation 23 Assumes that p t (x,v)→ 0 as ∥v∥→∞ is satisfied. 136 from (5.94), we get: Z T 0 Z 1 2 (∇ v · (p t ˆ a))dvdxdt =− Z T 0 Z 1 2 ∂ t p t (x,v)dvdxdt− Z T 0 Z 1 2 v·∇p t dvdxdt |z =0 =− Z T 0 Z 1 2 ∂ t p t logp t dvdxdt =− Z T 0 1 2 ∂ t Z p t logp t dvdxdt(5.99) DefiningF(t) := R p t logp t dvdxgives∂ t F(t) =∂ t R p t logp t dvdx, so we can apply the Fundamental Theorem of Calculus which states R T 0 F ′ (t)dt = F (T )− F (0) to get: Z T 0 Z 1 2 ⟨ ˆ a,∇ v logp t ⟩p t dxdvdt = Z 1 2 (p T logp T − p 0 logp 0 )dvdx(5.100) Finally, substituting this back into (5.96), we get the new form of the cost function with respect to ˆ a t : arg min a t Z T 0 Z 1 2 ∥a t ∥ 2 p t + σ 2 t 8 ∥∇ v logp t ∥ 2 p t dxdvdt + Z 1 2 (p T logp T − p 0 logp 0 )dxdv |z constant (5.101) In the SBM setting, the phase space marginals are constrained top 0 =π 0 andp T =π T , so we can drop the second term of the minimization, and write the multi-marginal SB problem as: a ⋆ t = arg min a t Z T 0 Z 1 2 ∥a t ∥ 2 p t + σ 2 t 8 ∥∇ v logp t ∥ 2 p t dxdvdt(5.102) s.t. ( ∂ t p t (x,v) +v·∇p t +∇ v · ( ˆ a t p t ) = 0 μ(x,t k ) = R R d π t k (x,v)dv, ∀k ∈0,...,K (5.103) This derivation shows that the multi-marginal Schr ̈odinger bridge problem in phase space can be reformulated as an entropy-regularized dynamic optimal transport problem over joint position–velocity distributions. The quadratic acceleration cost 1 2 ∥a t ∥ 2 governs how trajectories bend in phase space, penalizing deviations from inertial motion and enforcing smooth transitions that interpolate through the prescribed intermediate marginalsπ t k K k=1 . In addition, the Fisher information term σ 2 t 8 ∥∇ v logp t ∥ 2 acts specifically in the velocity coordinate, regularizing the uncertainty and dispersion of velocities induced by stochasticity. Just like how the static and dynamic Schr ̈odinger bridge problems are an entropically regularized analogue of static and dynamic optimal transport, the multi-marginal SB problem can be interpreted as the stochastic, entropically regularized analogue of measure-valued splines. This perspective will help us understand the behavior of multi-marginal SB in the zero-noise limit. Connection to Measure-Valued Splines The variational spline problem aims to select the smoothest curve (x(t)) t∈[0,T ] that interpolates between a set of sequential points ̄ x t 0 ,..., ̄ x t K ∈ R d at times 0 =t 0 <· < t k < 137 · < t K = T by minimizing: min (x t ) t∈[0,T ] Z T 0 ∥ ̈ x t ∥ 2 dt s.t. x t k = ̄ x t k ,∀k ∈0,...,K(5.104) where the minimization is taken over all twice-differentiable curves satisfying the interpolation constraints. The objective penalizes squared acceleration∥ ̈ x t ∥ 2 , so the resulting trajectory is the smoothest curve connecting the prescribed points in the sense of minimizing the total bending energy. This principle can be generalized from deterministic trajectories to evolving probability distributions by replacing the single curvex(t) with a time-dependent probability density p t (x,v) defined over the phase space of particle positions and velocities (x,v)∈ R 2d . Then, the distributional analogue of the variational spline problem, known as the measure-valued spline problem, becomes: inf a,p t Z T 0 Z R d Z R d ∥a(x,v,t)∥ 2 p t (x,v)dxdvdt (Measure-Valued Spline Problem) s.t. dx t = vdt, dv t = a t dt ∂ t p t +⟨v,∇p t ⟩ +∇ v · (ap t ) = 0 R R d p t (x,v)dv = π t k ,∀k ∈0,...,K where∂ t p t +⟨v,∇p t ⟩+∇ v ·(ap t ) = 0 is the phase-space continuity equation and R R d p t (x,v)dv= π t k enforces the position marginals. Intuitively, this problem yields the evolution of a proba- bility distribution that passes through prescribed marginal distributions while minimizing the average squared acceleration of the particles. Notice that this problem is exactly the (Multi-Marginal SB Problem) except with zero diffusion σ t ≡0, which reduces the Fokker-Planck equation with the Laplacian term σ 2 t 2 ∆ v p t ≡0 into the classic phase-space continuity equation. To this end, the multi-marginal SB problem can be interpreted as a stochastic, entropy-regularized analogue of the measure-valued spline. This formulation extends the dynamic Schr ̈odinger bridge problem along two key directions: in- corporating multiple intermediate marginal constraints and lifting the dynamics to phase space, where marginals can encode both position and velocity information. In the zero-noise limit, the problem recovers a deterministic measure-valued spline problem, revealing how SB generalizes smooth trajectory interpolation between distributions with stochastic uncertainty. 5.4 Unbalanced Schr ̈odinger Bridge Problem Since the Schr ̈odinger bridge problems originate from the optimal mass transport (OMT) problem, where the probability mass across the full time horizon is conserved, and no mass is lost. In many applications, such as cell dynamics, probability mass is not necessarily conserved, and particles undergo growth and death, producing unbalanced marginal distributions. However, this presents the key problem of determining the way in which particles should transport and vanish along intermediate time points that minimizes the deviation from some reference dynamics while reconstructing the unbalanced marginals. To account for the difference in mass between terminal marginals, we can relax the mass conservation constraint in the standard regularized OT problem by introducing a growth rateg(x,t) :R d × [0,T]→ Rinto the minimization objective to get the dynamic unbalanced optimal transport problem (Chizat et al., 2018a; b). 138 Figure 12: Unbalanced Schr ̈odinger Bridge Problem. The unbalanced SB problem extends the classical formulation to settings where the endpoint marginalsπ 0 andπ T have different total mass. In addition to the control u(x,t) that transports probability, the dynamics contain a growth termg(x,t) which allows mass to be created or removed along trajectories, which contributes a growth penalty Ψ(g) in the minimization objective. The dynamics of the density evolve according to the unbalanced Fokker–Planck equation, which augments the standard continuity equation with the mass-change term gp t . Definition 5.12 (Dynamic Unbalanced Optimal Transport Problem). Letπ 0 ,π T ∈M + (R d ) be non-negative measures that may have different total mass. The dynamic unbalanced optimal transport problem seeks a time-dependent densityp t , transport velocityv(x,t) : R d × [0,T ]→ R d , and growth rate g(x,t) : R d × [0,T ]→ R that solve the minimization: inf p t ,v,g Z T 0 Z R d 1 2 ∥v(x,t)∥ 2 + αΨ(g(x,t)) p t (x)dxdt(5.105) s.t. ( ∂ t p t (x) =−∇· (p t (x)v(x,t)) + g(x,t)p t (x) p 0 = π 0 , p T = π T (5.106) where Ψ :R→ R ≥0 is a non-negative function that penalizes changes in mass andα >0 is a hyperparameter that controls the penalty. Notice that the unbalanced continuity equation constraint in the dynamic unbalanced OT problem∂ t p t =−∇·(p t v) +gp t contains an additionalgp t term compared to the standard continuity equation. This term models local mass creation and destruction, where the current densityp t (x) at xgrows or decays proportionally tog(x,t). Therefore,gp t accounts for the instantaneous change in mass density due to growth or decay, allowing the total mass to vary over time. The growth penalty Ψ(g), commonly defined as the quadratic growth Ψ(g(x,t)) :=|g(x,t)| 2 , regularizes the amount of mass creation or destruction along the transport path. By penalizing large growth rates, the optimization balances mass transport and mass variation, yielding the minimal-cost combination of transport and local mass change required to match the terminal marginals. Extending this formulation to SDEs of the formdX t = (f(X t ,t) +σ t u(X t ,t))dt+σ t dB t yields the unbalanced Schr ̈odinger bridge problem, which seeks the most likely stochastic process between two unbalanced marginals (Chen et al., 2022; Pariset et al., 2023; Zhang et al., 2024). 139 Definition 5.13 (Unbalanced Schr ̈odinger Bridge Problem). Given a pair of unbalanced marginalsπ 0 ,π T ∈ P(R d ), a reference path measureQwith driftf(x,t) and a function that penalizes the growth rate Ψ :R → R ≥0 , the unbalanced Schr ̈odinger bridge (SB) problem aims to determine the optimal tuple density evolutionp ⋆ t , control driftu ⋆ , and growth rate g ⋆ that solve the minimization problem: inf p t ,u,g Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 + αΨ(g(x,t)) p t (x)dxdt(Unbalanced SB Problem) subject to the marginal constraintsp 0 =π 0 andp T =π T and an unbalanced Fokker-Planck constraint defined as: ∂ t p t (x) =−∇· (p t (x)(f (x,t) + σ t u(x,t))) + σ 2 t 2 ∆p t (x) + g(x,t)p t (x) (Unbalanced FP Equation) whereg(x,t)p t (x) relaxes the mass conservation constraint of the standard Fokker-Planck equation to allow the mass to change proportionally to the growth rate. Since the Schr ̈odinger bridge formulation has three optimization parameters, the densityp t , the control driftu, and the growth rategwhich are all coupled in the (Unbalanced FP Equation), optimizing a stochastic flow yields a difficult optimization problem. By reparameterizing the unbalanced Fokker- Planck constraint into a unbalanced continuity equation constraint, we can derive an alternative entropy-regularized dynamic optimal transport form of the (Unbalanced SB Problem), which was introduced to model stochastic unbalanced dynamical systems (Baradat et al., 2021; Buze et al., 2023; Chen et al., 2022; Janati et al., 2020), and has recently been used to develop scalable computational methods for simulating high-dimensional real-world data (Zhang et al., 2024). Proposition 5.14 (Entropy-Regularized Unbalanced Dynamic Optimal Transport Problem). The (Unbalanced SB Problem) from Definition 5.13 can be written as: inf (p t ,v,g) Z T 0 Z R d 1 2 ∥v∥ 2 + σ 2 t 8 ∥∇ logp t ∥ 2 − 1 2 ∇ logp t ,f − 1 2 (1 + logp t )g + αΨ(g) p t dxdt (Regularized Unbalanced OT) subject to the same marginal constraintsp 0 =π 0 andp T =π T and the unbalanced continuity equation constraint defined as: ∂ t p t (x) =−∇· (p t (x)(f (x,t) + σ t v(x,t)) + g(x,t)p t (x)(5.107) whereg(x,t)p t (x) relaxes the mass conservation constraint of the standard (Continuity Equation) a . a Note that the regularized unbalanced optimal transport objective introduced in (Zhang et al., 2024) is slightly different than the one stated here as we parameterize the control drift rather than the full velocity field, which results in an additional term containing the reference driftfand different coefficient scaling. Proof. Starting with the (Unbalanced FP Equation), we follow similar steps as in (Continuity Equation) to derive a reparameterized form of the control drift that satisfies a continuity equation 140 constraint with the additional growth term g(x,t)p t (x) as follows: ∂ t p t (x) =−∇· p t (x)(f (x,t) + σ t u(x,t)) + σ 2 t 2 ∆p t (x) + g(x,t)p t (x) =−∇· p t (x) f (x,t) + σ t u(x,t)− σ t 2 ∇ logp t (x) | z =:v(x,t) + g(x,t)p t (x) =−∇· (p t (x)(f (x,t) + σ t v(x,t)) + g(x,t)p t (x)(Unbalanced Continuity Equation) which is the unbalanced form of the Fokker-Planck constraint in (5.14). Definingv(x,t) :=u(x,t)− σ t 2 ∇ logp t (x) and rearranging to get the change-in-variablesu(x,t) =v(x,t) + σ t 2 ∇ logp t (x), we write the objective from Definition 5.13 as: inf (p t ,v,g) Z T 0 Z R d 1 2 ∥u(x,t)∥ 2 + αΨ(g(x,t)) p t (x)dxdt = inf (p t ,v,g) Z T 0 Z R d " 1 2 v(x,t) + σ t ∇ logp t (x) 2 + αΨ(g(x,t)) # p t (x)dxdt = inf (p t ,v,g) Z T 0 Z R d 1 2 ∥v(x,t)∥ + σ t ⟨v(x,t),∇ logp t (x)⟩ + σ 2 t 4 ∥∇ logp t (x)∥ 2 + αΨ(g(x,t)) p t (x)dxdt = inf (p t ,v,g) Z T 0 Z R d 1 2 ∥v(x,t)∥ 2 |z kinetic energy + σ t 2 ⟨v(x,t),∇ logp t (x)⟩ |z cross term + σ 2 t 8 ∥∇ logp t (x)∥ 2 |z Fisher information + αΨ(g(x,t)) |z growth penalty p t (x)dxdt (5.108) which follows the same steps as the derivation in Section 2.9. The expansion of the cross term deviates from the derivation in Section 2.9 as it incorporates the additional growth term in the continuity equation substitution: H(p T )− H(p 0 ) (2.93) = Z T 0 Z R d (1 + logp t )∂ t p t dxdt(5.109) Substituting the unbalanced continuity equation constraint∂ t p t =−∇·(p t (f+σ t v)) +gp t , we have: H(p T )− H(p 0 ) = Z T 0 Z R d (1 + logp t ) −∇· (p t (f + σ t v) + gp t dxdt = Z T 0 Z R d (1 + logp t ) −∇· (p t (f + σ t v) dxdt + Z T 0 Z R d (1 + logp t )(gp t )dxdt(5.110) Then, applying the integration of parts identity on the divergence and following the steps in (2.94), we get: H(p T )− H(p 0 ) = Z T 0 Z R d ∇ logp t ,p t (f + σ t v) dxdt + Z T 0 Z R d (1 + logp t )(gp t )dxdt = Z T 0 Z R d ∇ logp t ,f p t dxdt + Z T 0 Z R d σ t ∇ logp t ,v p t dxdt + Z T 0 Z R d (1 + logp t )(gp t )dxdt (5.111) Dividing both sides by two and rearranging to isolate the cross term in (5.108), we have: Z T 0 Z R d σ t 2 ∇ logp t ,v p t dxdt = 1 2 (H(p T )− H(p 0 )) + Z T 0 Z R d 1 2 − ∇ logp t ,f p t − (1 + logp t )(gp t ) dxdt = 1 2 (H(p T )− H(p 0 )) |z constant + Z T 0 Z R d − 1 2 ∇ logp t ,f − 1 2 (1 + logp t )g p t dxdt (5.112) 141 Figure 13: Branched Schr ̈odinger Bridge Problem. The branched Schr ̈odinger bridge generalizes the classical formulation to transport an initial distributionπ 0 toward multiple target marginalsπ T,k K k=1 . A primary trajectory evolves from the source distribution and probabilistically branches into secondary branched paths, each governed by its own control and growth dynamics to reach a different terminal distribution. The objective minimizes the weighted kinetic energy of the controls across all branches, producing a stochastic branching flow that connects a single source to multiple target modes. Since the entropy difference is fixed given the marginal constraintsp 0 =π 0 andp T =π T , the entropy difference term can be dropped, and we get: inf (p t ,u,g) Z T 0 Z R d 1 2 ∥v∥ 2 + σ 2 t 8 ∥∇ logp t ∥ 2 − 1 2 ∇ logp t ,f − 1 2 (1 + logp t )g + αΨ(g) p t dxdt(5.113) subject to the marginal constraintsp 0 =π 0 ,p T =π T , and the (Unbalanced FP Equation), which reformulates the unbalanced SB problem as a entropy-regularized dynamic OT problem. □ The reformulation removes the coupling betweenvand∇ logp t in the cross interaction term ⟨v(x,t),σ 2 t ∇ logp t (x)⟩, leading to a more numerically stable and computationally tractable objective (Zhang et al., 2024). We can also observe that since the terminal marginals are fixed, the entropy difference is constant, and the objective depends solely on the path-dependent cost. This results in a problem with three key components: the quadratic cost of the velocity field 1 2 ∥v∥ 2 , the Fisher information term that penalizes sharp changes in density, and the contribution of the growth rate determined by the growth penalty Ψ(g) and the entropy change caused by creating or destroying mass through (1 + logp t ). 5.5 Branched Schr ̈odinger Bridge Problem Orthogonal to the problem of matching multiple subsequent marginals along the temporal evolution of probability mass is the problem of matching a complex terminal marginal distributionπ T with multiple modesπ T =π T,1 ⊕·⊕ π T,K . In practice, fitting a standard Schr ̈odinger bridge to accurately transport density from an initial distributionπ 0 to a multi-modal distributionπ T suffers from several challenges. One of these challenges is mode collapse, in which the probability density is concentrated in only one or a few terminal modes, thereby missing the full distribution. To overcome this, one may increase 142 the number of particles simulated from the initial distribution to improve the likelihood of discovering each terminal mode; however, this increases computational cost and does not guarantee accurate reconstruction of all modes and their relative density weights. Furthermore, Schr ̈odinger bridges not only sample from the terminal distribution but also recover the energy-minimizing temporal bridge between the distributions. For multimodal target distributions, this optimal bridge can inform us about branching times and mass redistribution; however, it remains challenging to simulate effectively with standard SB methods. This is the motivation behind the Branched Schr ̈odinger Bridge Problem (Tang et al., 2026), which enables simulation of branching and mass redistribution over a stochastic bridge to reconstruct multi-modal target distributions. Since branching can be interpreted as the unbalanced flow of probability mass from a primary branch to multiple diverging trajectories, it can be formalized as the sum of Unbalanced Schr ̈odinger bridges, where mass is progressively depleted from the primary branch and fed into the diverging paths to multiple terminal modes. Definition 5.15 (Branched Schr ̈odinger Bridge Problem). Consider an initial distributionπ 0 and a terminal distribution withKdistinct modesπ T :=π T,1 ⊕·⊕ π T,K on some potential energy landscape defined by a state costc(x,t). Denoting the control drift of each branch as u k (x,t) K k=0 , the growth rate of each branch asg k (x,t) K k=0 , and the accumulated weight of each branch asw t,k K k=1 , the Branched SB problem seeks the optimal set of control and growth rates u ⋆ k ,g ⋆ k K k=0 that solve the following minimization: inf u k ,g k K k=0 Z T 0 E p t,0 1 2 ∥u 0 (X t,0 ,t)∥ 2 + c(X t,0 ,t) w t,0 +(5.114) K X k=1 E p t,k 1 2 ∥u k (X t,k ,t)∥ 2 + c(X t,k ,t) w t,k dt(Branched SB Problem) s.t. dX t,k = (f (X t,k ,t) + σ t u k (X t,k ,t))dt + σ t dB t X 0 ∼ π 0 , X T,k ∼ π T,k w 0,k = δ k=0 , w T,k = w ⋆ T,k where the weight of the primary branch is given byw t,0 = 1 + R t 0 g 0 (X s,0 ,s)dsand the weights of the K secondary branches is given by w t,k = R t 0 g k (X s,k ,s)ds. When considering a setting where the total mass across all branches is conserved, we can add the constraint P K k=0 w t,k = 1 for allt ∈[0,T] which enforces that the the growth rates sum to zero, i.e.,g 0 (X t,0 ,t) + P K k=1 g k (X t,k ,t) = 0. In this setting, all the mass that is lost from the primary branch, such that growth rate is negativeg 0 <0, is redistributed to the secondary branches, such that at least one branch has positive growth∃k, g k >0. To formulate a tractable form of the (Branched SB Problem), we can reframe the problem as a conditional stochastic optimal control (CondSOC) problem that can be tractably solved given a finite set of samples from an empirical initial distribution π 0 and a multi-modal terminal distribution π T . Proposition 5.16 (Branched Conditional Stochastic Optimal Control (Proposition 2 in Tang et al. (2026))). Define the endpoint conditioned density of each branch asp t,k (X t,k ) := E π 0,T,k [p t,k (X t,k |x 0 ,x T,k )], whereπ 0,T,k is the joint coupling between the initial distribution π 0 and thekth mode of the terminal distributionπ T,k . The set of optimal control drifts and growth termsu k ,g k K k=0 that solve the Branched SB problem defined in Definition 5.15 can be obtained by minimizing the sum of Unbalanced Conditional Stochastic Optimal Control 143 problems defined as: inf u k ,g k K k=0 E (x 0 ,x T,0 )∼π 0,T,0 Z T 0 E p t|0,T,0 1 2 ∥u 0 (X t,0 ,t)∥ 2 + c(X t,0 ,t) w t,0 dt(5.115) + K X k=1 E (x 0 ,x T,k )∼π 0,T,k Z T 0 E p t|0,T,k 1 2 ∥u k (X t,k ,t)∥ 2 + c(X t,k ,t) w t,k dt s.t. dX t,k = (f (X t,k ,t) + σ t u k (X t,k ,t))dt + σ t dB t X 0 = x 0 , X T,k = x T,k w 0,k = δ k=0 , w T,k = w ⋆ T,k where the weight of the primary branch is given byw t,0 = 1 + R t 0 g 0 (X s,0 ,s)dsand the weights of the K secondary branches is given by w t,k = R t 0 g k (X s,k ,s)ds. Proof. To prove that the branched CondSOC problem solves the branched SB problem in Definition 5.15, we start by defining each branch k as solving its own Unbalanced SB problem, defined as: inf u k ,g k Z T 0 E p t,k 1 2 ∥u k (X t,k )∥ 2 + c(X t,k ,t) w 0,k + Z t 0 g s,k (X s,k )ds dt(5.116) s.t. ( ∂ t p t,k =−∇· (p t,k (f + σ t u k )) + σ 2 t 2 ∆p t,k + g k p t,k p 0 = π 0 , p T,k = π T,k (5.117) Now, it suffices to show that the sum of unbalanced CondSOC problems satisfies the global Fokker- Planck equation of the density over all branches, defined as: ∂ t p t (X t ) =−∇· (p t (X t )(f + σ t u t )(X t ,t)) + σ 2 t 2 ∆p t (X t )(5.118) where the probability densityp t is defined as the weighted sum of the density at each branch at time tgiven byp t (X t ) := P K k=0 w t,k p t,k (X t ). To derive the Fokker-Planck equation ofp t , we start by differentiating with respect to t and applying the chain rule to get: ∂ t p t = ∂ t " K X k=0 w t,k p t,k # = K X k=0 w t,k (∂ t p t,k ) | z branched FP + (∂ t w t,k ) | z :=g k (X t ) p t,k = K X k=0 w t,k −∇· ((f + σ t u k )p t,k ) + σ 2 t 2 ∆p t,k + g k p t,k = K X k=0 − w t,k ∇· ((f + σ t u k )p t,k ) + w t,k σ 2 t 2 ∆p t,k + g k p t,k = K X k=0 (−w t,k ∇· ((f + σ t u k )p t,k )) |z divergence + K X k=0 w t,k σ 2 t 2 ∆p t,k | z diffusion + K X k=0 g k p t,k |z growth (5.119) 144 First, we rewrite the divergence term into the form of the full Fokker-Planck equation (5.118) by applying the linearity of the divergence operator: K X k=0 − w t,k ∇· ((f + σ t u k )p t,k ) =−∇· K X k=0 w t,k p t,k (f + σ t u k ) ! =−∇· f K X k=0 w t,k p t,k + σ t K X k=0 w t,k u k p t,k =−∇· f 1 p t K X k=0 w t,k p t,k | z =1 +σ t 1 p t K X k=0 w t,k u k p t,k | z :=u p t = −∇· ((f + σ t u)p t ) (5.120) where the final equality follows from multiplying and dividing byp t . By definingu:= 1 p t P K k=0 w t,k u k p t,k , we have that the total control drift is the mass-weighted average of the control drift for each branch. This is theoretically grounded, as under the global context, the control driftu k (X t ,t) of a particle X t =xalong a single branchkshould be scaled by its probability of being in branchk, given by p t,k (X t ), the weight of the particle itselfw t,k (X t ), normalized by the total probability of the particle over all branches p t (X t ). Next, we rewrite the diffusion term as the diffusion of the full branched Fokker-Planck equation by applying the linearity of the Laplacian operator: K X k=0 w t,k σ 2 t 2 ∆p t,k = σ 2 t 2 ∆ K X k=0 w t,k p t,k ! | z :=p t = σ 2 t 2 ∆p t (5.121) Finally, for the growth term in (5.119), we observe that it is simply the weighted sum of the growth over each branch P K k=0 g k p t,k and doesn’t alter the direction or motion of the particle along the brnached fields in the global context. Therefore, all three terms in (5.119) satisfy the global Fokker-Planck equation, and we finish the proof.□ This derivation provides insight into how branched Schr ̈odinger bridges behave under the global dynamics. We observe that even when each branch evolves via its own control and growth field, the overall system evolves as a single stochastic process whose probability density is a weighted superposition of all branches. From the perspective of the global Fokker-Planck constraint, branching does not introduce additional forces or discontinuities in the particle motion. Instead, it induces a mixture of control drifts, where the effective drift at any state is the probability-weighted average of the branch-specific controls. This solves the challenge of mode collapse because each mode is generated with its own control drift, constrained by a terminal weight to ensure that the correct probability mass and distribution are reconstructed. Furthermore, the potential energy functionc(x,t) that governs the system dynamics are minimized at the optimal branched mass redistribution, such that the optimal control and growth fieldsu ⋆ k ,g k K k=0 that solve the Branched SB problem in Definition 5.15 yields the branching trajectories and their relative weights such that they minimize the total energy required to reconstruct the terminal distribution. 5.6 Fractional Schr ̈odinger Bridge Problem Up to this point, we have considered only stochastic differential equations (SDEs) with standard Brownian motion (BM) processes that are Markov, where each increment of the SDE is independent 145 of all previous steps. This formulation is a specific design choice that ensures the tractability of the bridge solution and corresponding drift. While this choice is often a sufficient approximation, it ignores the effects of long-range temporal dependencies inherent in complex real-world systems. Fractional Brownian motion (fBM) is a generalization of standard BM to non-memoryless processes, where each increment is dependent on previous increments. This dependence is characterized by the Hurst index (H), which determines the roughness or pathwise regularity of the dynamics and the magnitude of long-range dependencies. A Hurst index ofH= 0.5 recovers the standard BM. Definition 5.17 (Fractional Brownian Motion (L ́evy, 1963)). Given standardd-dimensional Brownian motion (BM) (B t ∈ R d ) t∈[0,T ] , fractional BM (fBM) is a centered Gaussian process defined by integrating over the history of the BM: B H t := 1 Γ H + 1 2 Z t 0 (t− s) H− 1 2 dB s , t≥ 0(5.122) where Γ is the Gamma function and H ∈ (0, 1) is the Hurst index. We observe that the kernel (t−s) H− 1 2 injects time-dependent memory to the BM by weighting recent states (s≈ t) heavily and weighting distant states lightly. ForH > 1 2 , the increments are positively correlated, resulting in smoother trajectories where each subsequent increment is more likely to be close to the previous increment. ForH < 1 2 , the increments are negatively correlated, resulting in rougher trajectories where each subsequent increment aims to undo itself or revert back to the mean. When H = 1 2 , the kernel reduces to (t− s) H− 1 2 = 1 which recovers the standard BM process. Given the dependence on previous states, fBM is not Markov, resulting in an intractable drift for simulation. To overcome this, Markov approximations of fBM have been introduced (Daems et al., 2023; Harms et al., 2019), which approximate fBM by a weighted sum of Ornstein–Uhlenbeck (OU) processes, which are a class of Markov BM processes with a restoring force pulling it back toward the mean. Definition 5.18 (Markov Approximation of Fractional Brownian Motion (Daems et al., 2023; Harms et al., 2019)). Given ad-dimensional fractional Brownian motion (fBM) (B H t ) t∈[0,T ] , we define a Markov-approximate fBM (MA-fBM) ( b B H t ) t∈[0,T ] as: b B H t := K X k=1 ω k Y k t , ω 1 ,...,ω K ∈ R(5.123) where Y k t denote K Ornstein–Uhlenbeck (OU) processes of the form: Y k t := Z t 0 e −γ k (t−s) dB s , dY k t =−γ k Y k t dt + dB t , k = 1,...,K(5.124) whereγ k K k=1 are the mean reversion coefficients that determine how strongly the dynamics are pulled back to the mean. To approximate the fBM dynamics,γ k :=r k−n forr >1 and n= K+1 2 , which produces a log-uniformly spread time grid of fast-reverting and slow-reverting OU processes. Intuitively, each OU process contributes one exponential memory scale, where short memory modes are captured by strong mean-reversion OU processes (largeγ k ) and long memory modes are captured by weak mean-reversion OU processes (smallγ k ). To determine the optimal finite set of coefficients 146 ω k K k=1 for MA-fBM, one can directly minimize the expected squared error between the true fBM and the OU approximation over time: (ω 1 ,...,ω K ) = arg min ω 1 ,...,ω K ( Z T 0 E P B H t − b B H t 2 dt ) (5.125) which yields a L 2 (P) optimal approximation 24 . Since the fBM process is determined by the scaled fBM process, which we denote as c X t := √ ε b B H t , and theKOU processesY t := (Y 1 t ,...,Y K t ) that approximate b B t , we define the full reference process Q as Z := ( √ ε b B t ,Y t ), which follows the SDE: Q : dZ t = FZ t dt + σ t dB t (5.126) whereF ∈ R d(K+1)×d(K+1) is a block matrix encoding the linear coupling between the stateX t and the auxiliary OU processes Y t := (Y 1 t ,...,Y K t ) and σ t is the diffusion coefficient. However, we observe that the process c X t alone is not Markov, as there is no way to predict the next state c X t+h from c X t without knowledge of the OU processes, therefore only the full reference process Z t := ( c X t ,Y 1 t ,...,Y K t ) is Markov. UsingQas the reference Markov process, we can derive the form of the Markov fractional Brownian bridge, where we leverage Doob’sh-transfrom described in Section 4.4 to condition Z t on the endpoint marginals. Proposition 5.19 (Markov Approximation of Fractional Brownian Bridge (Nobis et al., 2025)). LetQbe the reference path measure induced by the Markov augmentationZ t|0,T of a scaled fractional Brownian motion, with reference dynamics: dZ t|0,T = FZ t|0,T dt + σ t dB t (5.127) Fixing endpointsX 0 =x 0 andX T =x T , then the corresponding bridge measure obtained by conditioningQon these endpoint constraints is defined by the Doobh-transform ofQwith h(z,t) :=S T|t (x T |Z t|0,T =z), and the Markov approximation of the fractional Brownian bridge satisfies the SDE: dZ t|0,T = (FZ t|0,T + σ 2 t ∇ z log S T|t (x T |Z t|0,T = z))dt + σ t dB t (5.128) Proof. To prove this, we leverage Doob’sh-Transform described in Section 4.4, which defines a endpoint conditioned functionh(z,t). Here, we defineh:R d(K+1) ×[0,T]→[0,1] as the endpoint probabilty of a state X T = x T under the augmented path measure S, given by: h(z,t) := S T|t (X T = x T |Z t = z)≡ S T|t (x T |z)(5.129) where S T|t (·|z) is the conditional density on X T given z. Now, we aim to show thatS T|t (x T |z) satisfies the properties of theh-function defined in Section 4.4, which allows us to directly write the SDE associated with the fractional Brownian bridge. First, we check that it satisfies the (Martingale Property) that defines a valid tilting function: h(z,t) = Z R d S t+∆t|t (Z t+∆t = ̃ z|Z t = z)h( ̃ z,t + ∆t)d ̃ z(5.130) 24 while this does not yield strong pathwise convergence like in (Harms, 2019), it is more computationally tractable 147 To derive an expression forS t+∆t|t ( ̃ z|z), we use Bayes rule to decomposeS t+∆t|t,T ( ̃ z|z,x T ) and apply the Markov property S T|t,t+∆t (x T |z, ̃ z) = S T|t+∆t (x T | ̃ z) to get: S t+∆t|t,T ( ̃ z|z,x T ) = S T|t,t+∆t (x T |z, ̃ z)S t+∆t|t ( ̃ z|z) S T|t (x T |z) = S T|t+∆t (x T | ̃ z)S t+∆t|t ( ̃ z|z) S T|t (x T |z) =⇒ S t+∆t|t ( ̃ z|z) = S t+∆t|t,T ( ̃ z|z,x T )S T|t (x T |z) S T|t+∆t (x T | ̃ z) = S t+∆t|t,T ( ̃ z|z,x T )h(z,t) h( ̃ z,t + ∆t) (5.131) where we substitute the definition ofh(z,t) andh( ̃ z,t + ∆t) in the final equality. Now, substituting (5.131) into (5.130), we get: h(z,t) = Z R d S t+∆t|t (Z t+∆t = ̃ z|Z t = z)h( ̃ z,t + ∆t)d ̃ z = Z R d S t+∆t|t,T ( ̃ z|z,x T )h(z,t) h( ̃ z,t + ∆t) h( ̃ z,t + ∆t)d ̃ z = h(z,t) Z R d S t+∆t|t,T ( ̃ z|z,x T )d ̃ z |z =1 = h(z,t)(5.132) which means thath(z,t) :=S T|t (x T |z) is a valid tilting function and by Proposition 4.7 and the reference SDE (5.126), the SDE of the fractional Brownian bridge can be written as: dZ t|0,T = (FZ t|0,T + σ 2 t ∇ z log S T|t (x T |Z t|0,T = z))dt + σ t dB t (5.133) which concludes the proof of the fractional Brownian bridge.□. We can derive the explicit form of∇ z log S T|t (x T |Z t|0,T =z), which turns out to be Gaussian, with the following Lemma. Lemma 5.20 (Gaussian Form of Fractional Brownian Bridge). The fractional Brownian bridge has a Gaussian transition densityS t+∆t|t (·|z). Conditioned on the terminal state c X T =x T , the gradient of the log density takes the form: ∇ z log S T|t (x T |z) = h ∇ z log S 1 T|t (x T |z),...,∇ z log S d T|t (x T |z) i (5.134) where for each i∈1,...,d, we have: ∇ z log S i T|t (x T |z) = [1,ω 1 ζ 1 (t,T ),...,ω K ζ K (t,T )] ⊤ x i T −μ i T|t (z) σ 2 T|t (5.135) where the conditional mean μ T|t and covariance σ 2 T|t are given by: μ T|t (z) = x + K X k=1 ω k y k ζ k (t,T ), σ 2 T|t (z) = ε K X k=1 K X ℓ=1 ω k ω ℓ γ k + γ ℓ 1− e −(1−t)(γ k +γ ℓ ) (5.136) where z = (y 1 ,...,y K ) are the states of the OU processes. Proof. First, we recall the definition of the Markov-approximated fractional BM process given by the state X t and the OU random variables (Y 1 t ,...,Y K t ) defined as: c X t := √ ε K X k=1 ωY k t , dY k t =−γ k Y k t dt + dB t (5.137) 148 which combines to give the expression for the time evolution d c X t as: d c X t = √ ε K X k=1 ω k dY k t = √ ε K X k=1 ω k (−γ k Y k t dt + dB t ) =− √ ε K X k=1 ω k γ k Y k t dt + √ ε K X k=1 ω k dB t (5.138) Now, we want to derive an expression for the next state c X t+∆t by taking the integral R t+∆t 0 to get: c X t+∆t := Z t+∆t 0 − √ ε K X k=1 ω k γ k Y k r dr + √ ε K X k=1 ω k dB r ! =− √ ε K X k=1 ω k γ k Z t+∆t 0 Y k r dr + √ ε K X k=1 ω k B t+∆t (5.139) Splitting the integral into R t+∆t 0 = R t 0 + R t+∆t t and the Brownian incrementB t+∆t =B t + (B t+∆t − B t ), we have: c X t+∆t = √ ε − K X k=1 ω k γ k Z t 0 Y k r dr + K X k=1 ω k B t ! | z = b B H t − √ ε K X k=1 ω k γ k Z t+∆t t Y k r dr + √ ε K X k=1 ω k (B t+∆t −B t ) (5.140) Expanding Y k r using the definition in (5.124), we have: Y k r = Z r 0 e −γ k (r−s) dB s = Z t 0 e −γ k (r−s) dB s + Z r t e −γ k (r−s) dB s = Z t 0 e −γ k (r−t) e −γ k (t−s) dB s + Z r t e −γ k (r−s) dB s = e −γ k (r−t) Z t 0 e −γ k (t−s) dB s | z =:Y k t + Z r t e −γ k (r−s) dB s = e −γ k (r−t) Y k t + Z r t e −γ k (r−s) dB s (5.141) Substituting this back into (5.140) and applying the Stochastic Fubini Theorem (Harms et al., 2019), we get: c X t+∆t = √ ε b B t + √ ε K X k=1 ω k −γ k Z t+∆t t e −γ k (r−t) Y k t + Z r t e −γ k (r−s) dB s dr + (B t+∆t −B t ) = √ ε b B t | z =: c X t + √ ε K X k=1 ω k − γ k Y k t Z t+∆t t e −γ k (r−t) dr |z = R ∆t 0 e −γ k s ds=γ −1 k (1−e −γ k ∆t ) + Z t+∆t t Z r t e −γ k (r−s) dB s dr | z R t+∆t t R r t (·)dB s dr= R t+∆t t R t+∆t s (·)drdB s + (B t+∆t −B t ) = c X t + √ ε K X k=1 ω k −(1− e −γ k ∆t )Y k t − γ k Z t+∆t t Z t+∆t s e −γ k (r−s) drdB s + Z t+∆t t dB s = c X t + √ ε K X k=1 ω k (e −γ k ∆t − 1)Y k t + γ k Z t+∆t t − Z t+∆t s e −γ k (r−s) dr | z =1−e −γ k (t+∆t−s) +1 dB s = c X t + √ ε K X k=1 ω k (e −γ k ∆t − 1)Y k t + γ k Z t+∆t t e −γ k (t+∆t−s) dB s = c X t + K X k=1 ω k Y k t √ ε(e −γ k ∆t − 1) | z =:ζ(t,t+∆t) + √ ε K X k=1 ω k γ k Z t+∆t t e −γ k (t+∆t−s) dB s = c X t + K X k=1 ω k Y k t ζ(t,t + ∆t) + √ ε K X k=1 ω k γ k Z t+∆t t e −γ k (t+∆t−s) dB s (5.142) 149 To derive the conditional mean μ T|t (z) and covariance σ 2 T|t , we set ∆t = T − t in (5.142) to get: c X T = c X t + K X k=1 ω k Y k t ζ(t,T ) + √ ε K X k=1 ω k γ k Z T t e −γ k (T−s) dB s (5.143) DefiningZ t := ( c X t ,Y 1 t ,...,Y K t ), the conditional mean given the realizationz= (x,y 1 ,...,y K ) is given by: μ T|t (z) := E[ c X t |Z t = z] = x + K X k=1 ω k y k ζ(t,T )(5.144) where the stochastic integral vanishes under the conditional expectation. To compute the conditional variance σ 2 T|t := Var( c X T |Z t ), we let I k = R T t e −γ k (T−s) ds and expand: σ 2 T|t := Var( c X T |Z t ) = Var √ ε K X k=1 ω k γ k I k ! = εVar K X k=1 ω k γ k I k ! = ε K X k=1 K X ℓ=1 ω k ω ℓ Cov (I k ,I ℓ )(5.145) Using Itˆo’s isometry, which states that the covariance of Itˆo integrals combines to: Cov (I k ,I ℓ ) = Cov Z T t e −γ k (T−s) ds, Z T t e −γ ℓ (T−s) ds ! = Z T t e −γ k (T−s) e −γ ℓ (T−s) ds = Z T t e −(γ k +γ ℓ )(T−s) ds = Z T−t 0 e −(γ k +γ ℓ )r dr = 1− e −(γ k +γ ℓ )(T−t) γ k + γ ℓ (5.146) We can derive in the final expression for the conditional covariance: σ 2 T|t = ε K X k=1 K X ℓ=1 ω k ω ℓ 1− e −(γ k +γ ℓ )(T−t) γ k + γ ℓ = ε K X k,ℓ=1 ω k ω ℓ γ k + γ ℓ 1− e −(γ k +γ ℓ )(T−t) (5.147) Therefore, we have shown that c X T |(Z t =z)∼ S T|t (x T |z) =N(μ T|t (z),σ 2 T|t ). We compute the gradient for the log density ∇ z log S i T|t (x T |z) for the ith dimension of x T as: ∇ z log S i T|t (x T |z) =∇ z log 1 q 2πσ 2 T|t exp − (x i T −μ i T|t (z)) 2 2σ 2 T|t ! =∇ z − (x i T −μ i T|t (z)) 2 2σ 2 T|t + C ! = x i T −μ i T|t (z) σ 2 T|t ∇ z μ i T|t (z)(5.148) where the last equality follows from applying the chain rule since onlyμ T|t (z) depends onz. Given z := (x,y 1 ,...,y K ) and our definition for μ T|t (z) in (5.144), the gradient expands into: ∇ z log S i T|t (x T |z) = x i T −μ T|t (z) σ 2 T|t ∇ z μ i T|t (z) = x i T −μ i T|t (z) σ 2 T|t " ∂μ i T|t ∂x i , ∂μ i T|t ∂y i 1 ,..., ∂μ i T|t ∂y i K # ⊤ = [1,ω 1 ζ 1 (t,T ),...,ω K ζ K (t,T )] ⊤ x i T −μ i T|t (z) σ 2 T|t (5.149) which is exactly the form for the log gradient of the h-function defined in the Lemma.□ 150 Now, we can substitute this into the SDE of the fractional Brownian bridge from (5.133) to get its complete form: dZ t|0,T = (FZ t|0,T + σ 2 t u(Z t|0,T ,t))dt + σ t dB t s.t. u(Z t|0,T ,t) = [u 1 (Z t|0,T ,t),...,u d (Z t|0,T ,t)] u i (Z t|0,T ,t) =∇ z log S i T|t (x T |z) = [1,ω 1 ζ 1 (t,T ),...,ω K ζ K (t,T )] ⊤ x i T −μ i T|t (z) σ 2 T|t (5.150) By leveraging the finite-dimensional Markov lift of the MA-fBM approximation, we have derived the explicit conditional law of c X T givenZ t , which remains Gaussian with affine mean and time-dependent variance and whoseh-function can be derived in closed form. The non-Markovian nature of the marginal processX t provides a principled way to incorporate long-range temporal correlations into generative stochastic dynamics through a finite-dimensional Markov approximation of MA-fBM. This framework therefore, extends classical Schr ̈odinger bridge problems beyond memoryless dynamics and offers a mathematically tractable route to learning physically realistic long-horizon dependencies. 5.7 Closing Remarks for Section 5 In this section, we expanded the theory of the dynamic Schr ̈odinger bridge problem to a broader array of constraints and problem settings. We start with the Gassian SB problem, which we show admits a closed-form solution that is in the class of Gaussian Markov processes. We then introduced the generalized SB problem, which extends the classical formulation to systems with mean-field interactions where the dynamics of individual particles depend on the evolving population distribution. Next, we explored several important extensions of the SB framework that arise in more complex settings. These include the multi-marginal SB problem, which incorporates multiple intermediate marginal constraints; the unbalanced SB problem, which allows for the creation or destruction of mass along the transport trajectory; and the branched SB problem, which captures scenarios where stochastic trajectories diverge toward multiple terminal modes. Finally, we considered an alternative class of stochastic processes driven by fractional Brownian motion, leading to the formulation of the fractional SB problem. These extensions illustrate the incredible flexibility of the Schr ̈odinger bridge framework in describing complex stochastic systems across a wide range of settings. Having introduced the theoretical foundations of both the static and dynamic formulations of the SB problem, in addition to their extensions to diverse constraints and dynamics, we are now prepared to dive into modern generative modeling frameworks that leverage Schr ̈odinger bridge theory to construct scalable algorithms for high-dimensional data. 151 6. Generative Modeling with Schr ̈odinger Bridges In this section, we develop the connection between Schr ̈odinger bridge theory and modern generative modeling frameworks, showing how generative modeling can be formulated as the problem of learning controlled stochastic dynamics that interpolate between an initial and a target distribution while minimizing relative entropy with respect to a reference process. We begin with a brief primer on score-based generative modeling (Section 6.1), highlighting its formulation in terms of forward and reverse-time stochastic processes. We then extend this perspective by jointly learning forward and backward controlled drifts through likelihood maximization over coupled forward–backward SDEs (Section 6.2). As an alternative paradigm, we introduce diffusion Schr ̈odinger bridge matching, which constructs generative models via path-space reciprocal and Markov projections with parameterized drifts (Section 6.3). Finally, we present two simulation-free approaches for learning Schr ̈odinger bridges: score and flow matching (Section 6.4) and adjoint matching (Section 6.5). Throughout this section, we show how Schr ̈odinger bridges provide a unifying framework that connects likelihood-based training, path-space KL minimization, and score and flow matching frameworks into a single coherent theory. 6.1 A Primer on Score-Based Generative Modeling From Section 4.2, we have shown that Schr ̈odinger bridges are a generalization of diffusion models where the prior distribution is a simple Gaussian prior. Recall the backward SDE (Time Reversal Formula) corresponding to the forward SDE with variance-exploding drift given by: dX t = f (X t ,t)dt + σ t dB t ,X 0 ∼ π 0 := p data (6.1) d ̃ X s = h −f ( ̃ X s ,T − s) + σ 2 T−s ∇ log ̃p s ( ̃ X s ) i ds + σ T−s d e B s , ̃ X 0 ∼ π T := p prior (6.2) where ̃p s (x) is the density generated by the diffusion SDE at timesand∇ log ̃p s (x) is a gradient drift that pushes the density towards areas of high likelihood given the noisy data distribution at times. This process guides the diffusion process to samples from the true data distribution ̃p T (x) =p 0 (x). The expression ∇ log ̃p s (x) is known as the score function. Score-based generative modeling aims to train a generative model that samples from the data distributionπ 0 :=p data by parameterizing the score function with a neural network with parameters θknown as the score-based models θ (x,s)≈∇ log ̃p s (x) which estimates the score function over the state space x∈ R d and time coordinate s∈ [0,T ]. Given the score-based model s θ (x,s), we can define the estimated backward SDE as: d ̃ X s = h −f ( ̃ X s ,T − s) + σ 2 T−s s θ ( ̃ X s ,s) i ds + σ T−s d e B s , ̃ X 0 ∼ p prior (6.3) The distribution of clean samples generated from simulating many samples ̃ X 0 ∼ p prior via the SDE (6.3) over time s∈ [0,T ] approximates the data distribution p data . A simple objective that is minimized exactly when the score-based model matches the true score function is the score matching lossL SM (Hyv ̈arinen et al., 2005; Song et al., 2021) defined as the weighted squared difference integrated over time: L SM (θ) := 1 2 Z T 0 E ̃p s λ(t)∥∇ log ̃p s (x)−s θ (x,s)∥ 2 ds(Score Maching Loss) For this objective to be tractable, we require a closed-form expression for∇ log ̃p s (x). For generative modeling of a clean data distribution, the forward stochastic process or noise injection process can 152 Figure 14: Score-Based Generative Models. Score-based generative models learn the score function∇ logp t (x) of a diffusion process that progressively perturbs data with noise through a forward SDE. A neural networks θ (x,t) is trained using the score matching loss to approximate this time-dependent score. Once learned, the reverse-time SDE uses the estimated score to iteratively denoise samples, transforming noise into data. be defined with the conditional distributionp t ( ̃ x t |x 0 ) :=N( ̃ x t ;x 0 ,σ 2 t I d ) wherex 0 ∼ p data , which progressively smoothes the data distribution with larger varianceσ t > σ s fort > s(Song et al., 2019). This yields the tractable score function: ∇ logq t ( ̃ x t |x 0 ) =∇ log 1 p 2πσ 2 t exp − ∥ ̃ x t −x 0 ∥ 2 2σ 2 t =∇ − ∥ ̃ x t −x 0 ∥ 2 2σ 2 t + C = x 0 − ̃ x t σ 2 t (6.4) which can be optimized with the score matching objective: L SM (θ) := 1 2 Z T 0 E p t ( ̃ x t |x 0 ),p data (x 0 ) " x 0 − ̃ x t σ 2 t −s θ (x,t) 2 # dt(6.5) While score matching provides a simulation-free way to learn the reverse-time drift through estimation of the score function, it does so under the assumption that the generative process has a simple prior distribution, typically a standard Gaussian like in (6.4), and the forward diffusion dynamics are chosen to be linear like in (6.5) such that it is analytically tractable without simulation. This design constraint restricts the class of admissible dynamics to perturbations of a simple reference process which transforms between noise and data. To extend this idea to model transport between structured distributions with an unknown forward control drift, we introduce a likelihood-based training framework for the forward-backward SDEs that characterize the Schr ̈odinger bridge. 6.2 Likelihood Training of Forward-Backward SDEs Prerequisite: Section 4.3 Just like how likelihood training provides a theoretically-grounded training objective of estimating the score function in score-based generative modeling, we can derive a lower bound for the log-likelihood of the forward-backward stochastic differential equations (SDEs) defined in Section 4.3, which yields a tractable training objective that aims to maximize the lower bound (Chen et al., 2021). Since the definition ofY t =logφ t (X t ) and b Y t =log ˆφ t (X t ) from Section 4.3 jointly determine the log density of the SB solutionY t + b Y t =logp ⋆ t (X t ), we can write the likelihood under the FBSDEs 153 Figure 15: Likelihood Training of Forward and Backward Schr ̈odinger Potentials. The Schr ̈odinger bridge can be learned by parameterizing the log gradient of the forward and backward potentialsZ θ t =∇ logφ t (X t ) and b Z φ t =∇ log ˆφ t (X t ), which define the forward and backward SDEs between endpoint distributions.θandφare trained by maximizing path likelihoods through forward and backward objective functions, each evaluated under trajectories generated by the corresponding controlled SDE. Together, the two likelihood losses couple the forward and reverse dynamics, enabling the model to learn a consistent Schr ̈odinger bridge between the marginals. as the estimated value of Y 0 + b Y 0 = logp ⋆ 0 (X 0 ) given a data point X 0 = x 0 . logp ⋆ 0 (x 0 ) = E [logp ⋆ 0 (X 0 )|X 0 = x 0 ] = E h Y 0 + b Y 0 X 0 = x 0 i (6.6) First, we recall the set of three coupled FBSDEs that define the solution to the non-linear SB problem as: dX t = (f (X t ,t) + σ t Z t )dt + σ t dB t dY t = 1 2 ∥Z t ∥ 2 dt +Z ⊤ t dB t d b Y t = ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 + σ 2 t Z ⊤ t b Z t dt + b Z ⊤ t dB t (6.7) Since theY t and b Y t evolve via backward SDEs with boundary constraintsY T + b Y T =logπ T (X T ), we can writeY 0 and b Y 0 as the terminal condition integrated backward in time via the SDEs defined in (4.50) such that: Y T −Y 0 = Z T 0 1 2 ∥Z t ∥ 2 dt + Z T 0 Z ⊤ t dB t (6.8) b Y T − b Y 0 = Z T 0 ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 + σ 2 t Z ⊤ t b Z t dt + Z T 0 b Z ⊤ t dB t (6.9) which we rearrange to get: Y 0 = Y T − Z T 0 1 2 ∥Z t ∥ 2 dt− Z T 0 Z ⊤ t dB t (6.10) b Y 0 = b Y T − Z T 0 ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 + σ 2 t Z ⊤ t b Z t dt− Z T 0 b Z ⊤ t dB t (6.11) 154 Substituting this into the log-likelihood, we get: logp ⋆ t (x 0 ) = E h Y 0 + b Y 0 X 0 = x 0 i = E Y T − Z T 0 1 2 ∥Z t ∥ 2 dt− Z T 0 Z ⊤ t dB t + b Y T − Z T 0 ∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 + σ 2 t Z ⊤ t b Z t dt− Z T 0 b Z ⊤ t dB t X 0 = x 0 (6.12) Combining similar terms, we get: logp ⋆ (x 0 , 0) = E h Y T + b Y T X 0 = x 0 i − Z T 0 E 1 2 ∥Z t ∥ 2 +∇· (σ t b Z t −f ) + 1 2 ∥ b Z t ∥ 2 + σ 2 t Z ⊤ t b Z t X 0 = x 0 dt − E Z T 0 Z ⊤ t dB t + Z T 0 b Z ⊤ t dB t X 0 = x 0 | z =0 (Itˆo integral has zero-expectation) (6.13) Since the Itˆo integral has zero-expectation and Y T + b Y T = logp T (X T ), we are left with: logp ⋆ 0 (x 0 ) = E h logp ⋆ ( X T ) i |z (i) − Z T 0 E 1 2 ∥Z t ∥ 2 | z (i) + 1 2 ∥ b Z t ∥ 2 | z (i) +∇· (σ t b Z t −f ) | z (iv) +σ 2 t Z ⊤ t b Z t |z (v) dt(6.14) which is the log-likelihood of the data point that we aim to maximize. Each term can be interpreted as follows: (i)The terminal distribution matching reward that is maximized with the distribution at timeT generated by the forward bridge matches p T . (i)The energy cost of steering samples via the forward potential driftZ t =σ t ∇ logφ t (X t ), which is minimized when the log-likelihood is maximized. (i)The energy cost of steering samples via the backward potential drift b Z t =σ t ∇ log ˆφ t (X t ), which is minimized when the log-likelihood is maximized. (iv)This divergence term ensures that the drift is consistent with the time-evolving density p ⋆ t (X t ). We can observe this by expanding the divergence term∇·(σ t b Z t ) using the identity ∇· (uv) = (∇u)·v + u∇·v rearranged to u∇·v =∇· (uv)− (∇u)·v: E p ⋆ t h ∇· (σ t b Z t ) i = Z R d σ t ∇· b Z t p ⋆ t dX t = Z R d ∇· (σ t p ⋆ t b Z t )dX t | z =0 − Z R d b Z t ·∇p ⋆ t dX t =− Z R d b Z t ·∇p ⋆ t dX t (6.15) (v) The consistency of the forward-backward potentials. Since we want the drifts to generate exactly symmetric opposite bridges and the dot product is minimized at−1 when the vectors are exactly opposite, this term is minimized when the log-likelihood is maximized. SinceZ t and b Z t determine the optimal drift in the forward and backward directions based on the SB potential, we can train a generative SB model by parameterizing them with neural networksZ θ t ≈ Z t 155 and b Z φ t ≈ b Z t with parametersθandφ. Defining the SB loss with (6.14), we get a theoretically- grounded maximization objective that lower bounds the true log-likelihoodlogp ⋆ 0 (x 0 )≥ L SB defined as: L SB (θ,φ) = E [logp ⋆ T (X T )]− Z T 0 E 1 2 ∥Z θ t ∥ 2 + 1 2 ∥ b Z φ t ∥ 2 +∇· (σ t b Z φ t −f ) + σ 2 t Z θ⊤ t b Z φ t dt (6.16) Jointly training bothZ θ t and b Z φ t with this objective is carried out by (i) simulating the forward trajectory of the SDE.X 0:T usingZ θ t and b Z φ t , (i) computing the maximum likelihood objectiveL SB with (6.16), and (i) backpropagating through the SDE solver for every time step with respect to both θ and φ, which requires maintaining the full computational graph of the SDE. While this training scheme works for low-dimensional data, it becomes computationally infeasible for high-dimensional data like images. To overcome this, we can store SDE trajectories in a replay buffer as sequences of static statesX t while discarding the computational graph containing the gradient path. Crucially, this breaks the dependency of the trajectories with the current model parameters, but it enables us to reuse the same SDEs over multiple gradient updates. Updating one model, likeZ θ t , with respect to∇ θ L SB would change the SDEX t , so updating b Z φ t simultaneously with the original SDE would break the symmetry of the optimization problem. Instead, we can leverage the symmetric property of Schr ̈odinger bridges: (i)Given the true forward potential driftZ t =σ t ∇ logφ t , the optimal backward potential drift b Z φ t can be learned to match the forward trajectories. (i)Given the true backward potential drift b Z t =σ t ∇ log ˆφ t , the optimal forward potential drift Z θ t can be learned to match the backward trajectories. To train the backward potential drift given trajectories generated with the frozen forward modelZ ̄ θ t , where ̄ θ:=stopgrad(θ), we can use the same likelihood maximization objective defined in (6.16) but dropping all terms that are not dependent on φ: ̃ L SB (φ) =− Z T 0 E X 0:T ∼P ̄ θ 1 2 ∥ b Z φ t ∥ 2 +∇· (σ t b Z φ t −f ) + σ 2 t Z θ⊤ t b Z φ t dt(6.17) s.t. P ̄ θ : dX t = (f + σ 2 t ∇ logφ t (X t ))dt + σ t dB t , X 0 ∼ π 0 (6.18) Given that the SB is symmetric in either direction, the objective for training the forward potential drift given trajectories generated with the frozen backward model b Z ̄ φ t , where ̄ φ :=stopgrad(φ), is the same as (6.17) but flipping Z θ t and b Z φ t given by: ̃ L SB (θ) =− Z T 0 E X 0:T ∼P ̄ φ 1 2 ∥Z θ t ∥ 2 +∇· (σ t Z θ t −f ) + σ 2 t b Z φ⊤ t Z θ t dt(6.19) s.t. P ̄ φ : dX t = (f − σ 2 t ∇ log ˆφ t (X t ))dt + σ t dB t , X T ∼ π T (6.20) Derivation sketch. This symmetric objective can be derived by defining a reversed time coordinate s:=T − tand defining the prior non-linear process as the reverse-time analog of the forward prior dynamics with SDEdX s =−f(X s ,s)ds+σ t dB s . Then, redefining the Hopf-Cole linear PDE constraints and forward-backward SDEs with negative drift−fand following a similar derivation from (6.8) to (6.14) for the log-likelihoodlogp ⋆ 0 (x T ) =E[Y 0 + b Y 0 |X T =x T ] of a samplex T from the prior distribution or π T .□ 156 This section shows that the optimal forward and backward control drifts defined by the Schr ̈odinger potentials can be learned through mazximizing a lower bound on the log-likelihood of reconstructing both of the marginal constraints. These objectives lead to a symmetric alternating optimization procedure that mirrors the structure of Sinkhorn’s algorithm from Section 1.5. Next, we move on to an alternative perspective, considering the optimization process as performing iterative Markovian and reciprocal projections rather than maximizing likelihoods. 6.3 Diffusion Schr ̈odinger Bridge Matching Prerequisite: Section 4.5 Building on the Iterative Markovian Fitting (IMF) procedure from Section 4.5, we now describe the Diffusion Schr ̈odinger Bridge Matching (DSBM) algorithm (Shi et al., 2023). This algorithm unifies ideas from denoising diffusion (Ho et al., 2020; Song et al., 2020b) and flow matching (Lipman et al., 2022) to solve the SB problem with arbitrary marginal distributions by parameterizing the Markov drift optimized to match the Schr ̈odinger bridge drift through Markovian and reciprocal projections. While the IMF procedure provides a theoretically grounded procedure for constructing the Schr ̈odinger bridge through alternating KL projections in path space, its formulation remains abstract and infinite- dimensional. To make these ideas computationally tractable in high-dimensional settings, we can leverage the explicit SDE representation of the Markovian projection, defined as: dX t = f (X t ,t) + σ 2 t E Π T|t ∇ log Q T|t (X T |X t )|X t dt + σ t dB t (Forward Markovian Projection SDE) and parameterize the forward-time Markov control driftσ t E Π T|t ∇ log Q T|t (X T |X t )|X t with u θ (x,t) such that it converges to the optimal drift u ⋆ through the sequence of IMF iterations. Although the Markovian projection preserves the bridge measureM ⋆ t = Π t in theory (Proposition 4.10), parameterizing only the forward-time SDE results in errors in practice, where the terminal marginalp T generated from simulating (Forward Markovian Projection SDE) may not exactly match the true marginal constraintπ T . Therefore, to avoid error accumulation during the IMF sequence, we also parameterize the reverse-time Markovian projection, which we define below. Proposition 6.1 (Forward and Reverse Time Markovian Projections (Proposition 9 in Shi et al. (2023)). Given a mixture of bridges Π = Π 0,T Q ·|0,T in the reciprocal class Π∈ R(Q) of the reference measureQgenerated by the SDEdX t =f(X t ,t)dt+σ t dB t , the Markovian projectionM:=proj M (Π) can be written as both forward and reverse time SDEs defined as: dX t = f (X t ,t) + σ 2 t E Π T|t ∇ log Q T|t (X T |X t )|X t dt + σ t dB t (6.21) d ̃ X s = −f (X s ,T − s) + σ 2 T−s E Π 0|T−s ∇ log Q T−s|0 ( ̃ X s | ̃ X T )| ̃ X s dt + σ T−s d ̃ B s (6.22) with initial conditions X 0 ∼ Π 0 and ̃ X 0 ∼ Π T , respectively. Proof. The proof of this proposition follows directly from the definition of the Markovian projection in (4.10) and applying the (Time Reversal Formula) described in Section 4.2.□ Parameterizing both the forward-time Markovian projection with driftu θ (x,t) and the reverse-time Markovian projection with driftu φ (x,t), we outline the Diffusion Schr ̈odinger Bridge Matching (DSBM) algorithm (Shi et al., 2023) as follows. 157 Diffusion Schr ̈odinger Bridge Matching DSBM generates a sequence of Markov projections (M n ) n∈N and reciprocal projections (Π n ) n∈N initialized at Π 0 := π 0,T Q ·|0,T by alternating between the following steps: (1a) Solve the forward-time Markovian projectionM 2n+1 :=proj M (Π 2n ) by updating a parameterized drift u θ to minimize L DSBM (θ) := KL(Π 2n ∥M θ ). (1b) Define the reciprocal projection as Π 2n+1 := M 2n+1 Q ·|0,T (2a)Solve the backward-time Markovian projectionM 2n+2 :=proj M (Π 2n+1 ) by updating a parameterized drift u φ to minimize L DSBM (φ) := KL(Π 2n+1 ∥M φ ). (2b) Define the reciprocal projection as Π 2n+2 := M 2n+2 Q ·|0,T To learn the forward and reverse time Markovian projections in Steps (1a) and (2a), we can minimize loss functions defined as the KL divergence 25 as derived in Section 2.6: L DSBM (θ) = Z T 0 E Π t,T σ t ∇ log Q T|t (X T |X t )−u θ (X t ,t) 2 dt(Forward DSBM Loss) L DSBM (φ) = Z T 0 E Π t,0 σ t ∇ log Q t|0 (X t |X 0 )−u φ (X t ,t) 2 dt(Reverse DSBM Loss) Given sufficient expressivity ofθ,φ, it is easy to see that optimizing the above losses for all (x,t) exactly yields the control drift of the Markov projection proj M (Π): u θ ⋆ (x,t) = σ t E Π T|t ∇ log Q T|t (X T |X t )|X t = x (6.23) u φ ⋆ (x,t) = σ t E Π 0|t ∇ log Q t|0 (X t |X 0 )|X t = x (6.24) The corresponding reciprocal projections performed in (1b) and (2b) are obtained by first simulating trajectoriesX 0:T either with the forward SDE usingu θ or in the reverse SDE usingu φ to obtain samples from the endpoint law (x 0 ,x T )∼ M 0,T , and then sampling from the conditional bridge X t ∼ Q(·|x 0 ,x T ) of the reference process Q. Optimizing these losses through Algorithm 6.3 exactly performs the Iterative Markovian Fitting (IMF) procedure from Section 4.5 through parameterized control drifts of SDEs. Therefore, by Theorem 4.17, we have that the unique fixed point of the diffusion SBM algorithm yields the optimal Schr ̈odinger bridgeP ⋆ . Rather than separating the forward and reverse Markovian projection steps, Shi et al. (2023) shows that they can be performed simultaneously, which mimics the true IMF procedure where each projection corresponds to the exact Markov and reciprocal projections. Joint Training of Forward and Reverse Markovian Projections Since both objectives can be computed from sampling a pair (x 0 ,x T )∼Π 0,T from the joint distribution and sampling the intermediate stateX t ∼ Q t|0,T (·|x 0 ,x T ) from the bridge measure, we can optimize bothθandφjointly. After each iterationn, the updated parameters can be used to define the forward Markov processM n+1 f and the backward Markov process M n+1 b with the following SDEs: M n+1 f : dX t = [f (X t ,t) + σ t u θ (X t ,t)]dt + σ t dB t ,X 0 ∼ π 0 (6.25) M n+1 b : d ̃ X s = [−f ( ̃ X,T − s) + σ T−s u φ ( ̃ X s ,T − s)]ds + σ T−s dB s ̃ X 0 ∼ π T (6.26) 25 In Shi et al. (2023), the parameterized control drift is not scaled by the diffusion coefficient, which yields an extra σ 2 t in the denominator. Since we define the control as σ t u θ in the SDEs, the extra σ 2 t vanishes. 158 Figure 16: Diffusion Schr ̈odinger Bridge Matching. Illustration of the diffusion Schr ̈odinger bridge matching (DSBM) procedure with alternating Markovian and reciprocal projections. Starting from an initial reciprocal measure Π 0 =π 0,T Q ·|0,T , the algorithm alternates between (i) forward-time Markovian projectionsM 2n+1 which updates the parameterized forward driftu θ to match Π 2n and (i) reverse-time Markovian projectionsM 2n+2 which updates the parameterized reverse driftu φ to match Π 2n+1 . Each Markovian projection is followed by a reciprocal projection that constructs a mixture of reference bridges conditioned on the updated endpoint laws, denoted Π 2n+1 and Π 2n+2 . This alternating procedure produces a sequence of path measures that progressively reduce the KL divergence to the optimal bridge, converging to the Schr ̈odinger bridge solution which occurs when the Markov and reciprocal projections reach equilibrium M ⋆ = Π ⋆ = P ⋆ . At equilibrium, the forward and backward SDEs should match, which means reversing M n f yieldsM n b and reversingM n b yields theM n f . To enforce this during training, we can compute the time-reversal of the backward SDE (6.26) using the (Time Reversal Formula) to get: dX t = [f (X t ,t) + σ t (−u φ (X t ,t) + σ t ∇ log Π 2n t (X t ) | z should match u θ (X t ,t) )]dt + σ t dB t , X 0 ∼ π 0 (6.27) To ensure the control drifts in (6.27) and (6.25) are aligned, we leverage a consistency loss defined as: L cons (θ,φ) = Z T 0 E Π 2n t ∥u θ (X t ,t) +u φ (X t ,t)− σ t ∇ log Π 2n t (X t )∥ 2 dt(Consistency Loss) = Z T 0 E Π 2n t u θ (X t ,t) +u φ (X t ,t)− σ t E Π 2n T|t ∇ log Q T|t |x + E Π 2n 0|t ∇ log Q t|0 |x 2 dt where we rewrite the score function∇ logΠ 2n t (X t ) with the known conditional densitiesQ T|t and Q 0|t using the identity: ∇ log Π 2n t (x) = E Π 2n T|t ∇ log Q T|t |X t = x + E Π 2n 0|t ∇ log Q t|0 |X t = x (6.28) which defines the marginal score at timetas the sum of the conditional Gaussian scores at the endpoints. Therefore, the total joint training loss can be defined as: L DSBM (θ,φ) =L DSBM (θ) +L DSBM (φ) + λL cons (θ,φ)(6.29) where λ > 0 is a positive weight that defines the strength of the consistency loss. This formulation provides a tractable approach for performing path-space projections using parame- terized drifts that can be simulated at inference time via an SDE solver. In practice, the Markov 159 Figure 17: Conditional Score and Flow Matching. Conditional score and flow matching learn the dynamics of stochastic bridges by conditioning trajectories on endpoint pairs (x 0 ,x T )∼ π 0,T . The model jointly estimates the conditional control driftu(x,t|x 0 ,x T ) and the conditional score∇ logp t (x|x 0 ,x T ), which describe how trajectories evolve between the endpoints. By minimizing the combined objective, the learned dynamics produce the mixture of Brownian bridges which is equal to the Schr ̈odinger bridge P ⋆ given the entropic OT coupling π ⋆ 0,T . projection requires learning both forward and backward drifts to satisfy the marginal constraints, despite the underlying equivalence of the bridge measures. However, this approach remains com- putationally intensive, as training still relies on simulating full stochastic trajectories due to the absence of a closed-form sampling procedure for the optimal bridge. Motivated by this limitation, we now consider an alternative perspective in which the intermediate states admit tractable expression, enabling simulation-free matching objectives. 6.4 Simulation-Free Score and Flow Matching Prerequisite: Section 4.1, 4.2 To overcome the restriction to the Gaussian prior distribution of score-based generative modeling described in Section 6.1, we highlight the simulation-free score and flow matching ([SF] 2 M; Tong et al. (2023b)) framework which extends score matching to arbitrary prior distributions. Crucially, we will show that given the optimal entropic OT coupling (Section Static SB Problem), [SF] 2 M solves the (Dynamic SB Problem) through a endpoint-conditioned objective with the same gradient as the unconditional objective. This framework considers a data-driven SB problem, where we have empirical samples from both the marginal distributionsx 0 ∼ π 0 andx T ∼ π T . In this setting, we can solve the entropic optimal transport (OT) problem between empirical samples to determine the optimal couplingπ ⋆ 0,T , from which the solution to the dynamic SB problem with Brownian reference processQis defined simply as a mixture of Brownian bridges weighted by the optimal static couplingπ ⋆ 0,T as proven in Proposition 4.3 and Corollary 4.2. Recall from Section 4.2, where we derived the (4.40) as: ( dX t = σ t u(X t ,t)dt + σ t dB t d ̃ X s =−σ t u( ̃ X s ,T − s) + σ 2 T−s ∇ logp( ̃ X s ,T − s)ds + σ T−s d e B s (6.30) where we define the reference drift as pure Brownian motionf:= 0. Crucially, the score function ∇ logp t (X t ) appears in the reverse-time drift as a correction term that compensates for the entropy- producing forward diffusion, ensuring that the reversed dynamics reproduce the correct marginal distributions. This decomposition of the control and score function motivates a combined objective 160 that learns a parameterized control driftu θ , which learns the reverse controlu θ (x,t)≈ u(x,t) 26 , and the score function ∇ logp t (x). Definition 6.2 (Unconditional Score and Flow Matching Objective). The unconditional score and flow matching objective aims to match a parameterized control fieldu θ (x,t) : R d ×[0,T]→ R d and score functions θ (x,t) :R d ×[0,T]→ R d to the true velocity and score function defining the solution to (Dynamic SB Problem) by minimizing: L U[SF] 2 M (θ) := Z T 0 E p t ∥u θ (x,t)−u(x,t)∥ 2 |z flow matching loss +λ(t) 2 ∥s θ (x,t)−∇ logp ⋆ t (x)∥ 2 |z score matching loss dt (Unconditional Objective) wherep ⋆ t is the optimal marginal density of the dynamic SB, andλ(t) : [0,T]→ Ris some positive weight. While this objective is theoretically sound, bothu(x,t) and∇ logp t (x) are undefined or intractable for general target distributions. In this setting, we assume access to explicit samples from both marginal distributionsπ 0 andπ T and define the tractable control drift and score for a Brownian bridge between a predefined coupling (x 0 ,x T )∼ π 0,T given by: u(x,t|x 0 ,x T ) = 1− 2t t(1− t) (x− (tx 1 + (1− t)x 0 )) + (x 1 −x 0 ) ∇ logp t (x|x 0 ,x T ) = tx 1 + (1− t)x 0 −x σ 2 t t(1− t) (6.31) Using this definition of the endpoint-conditioned velocity and score function, we can define the conditional velocity and score over the empirical distribution π 0 by taking an expectation: u(x,t) = E π 0,T u(x,t|x 0 ,x T )p t (x|x 0 ,x T ) p t (x) (6.32) ∇ logp t (x) = E π 0,T p t (x|x 0 ,x T ) p t (x) ∇ logp t (x|x 0 ,x T ) (6.33) These tractable definitions for the conditional control drift and score function motivate the definition of the conditional score and flow matching objective (Tong et al., 2023b), which we show yields the same gradients as the unconditional objective. Proposition 6.3 (Conditional Score and Flow Matching Objective (Theorem 3.2 in Tong et al. (2023b))). Consider the conditional score and flow matching objective which aims to match a parameterized control driftv θ (x,t) :R d ×[0,T]→ R d and score function s θ (x,t) :R d ×[0,T]→ R d to a distribution of velocity and score functions conditioned on the endpoint z ∼ π T by minimizing: L [SF] 2 M (θ) := Z T 0 E p t|0,T ,π 0,T ∥v θ (x,t)−u(x,t|x 0 ,x T )∥ 2 | z conditional flow matching loss +λ(t) 2 ∥s θ (x,t)−∇ logp t (x|x 0 ,x T )∥ 2 |z conditional score matching loss dt (Conditional Objective) where the expectation is taken over samples from the endpoint law (x 0 ,x T )∼ π 0,T and samples from the conditional distribution at timet ∈[0,T] given target endpointsx ∼ p t|0,T (·|x 0 ,x T ). Then, we have that the gradients of the conditional objective match the gradients of (Unconditional Objective) such that ∇ θ L [SF] 2 M (θ) =∇ θ L U[SF] 2 M (θ). 26 since the control is non-deterministic, we will just consider the reverse control asu(x,t) and omit the negative sign for simplicity. 161 Proof. The goal of this proof is to show the equivalence between the gradients of the conditional and unconditional expectations: ∇ θ E p t|0,T ,π 0,T ∥w θ (x,t)−w(x,t|x 0 ,x T )∥ 2 =∇ θ E p t ∥w θ (x,t)−w(x,t)∥ 2 (6.34) which can be applied for both the conditional flow matching loss withw(x,t|x 0 ,x T ) :=u(x,t|x 0 ,x T ) and the conditional score matching lossw(x,t|x 0 ,x T ) :=∇ logp t (x|x 0 ,x T ). Expanding the squared loss in the conditional objective, we have: ∥w θ (x,t)−w(x,t|x 0 ,x T )∥ 2 = ∥w θ (x,t)∥ 2 | z independent of x 0 ,x T −2⟨w θ (x,t),w(x,t|x 0 ,x T )⟩ +∥w(x,t|x 0 ,x T )∥ 2 |z independent of θ (6.35) where we observe that the first term is independent of the conditional pair (x 0 ,x T ), which means that it is clearly equivalent to the unconditional gradient. The last term is independent ofθwhich has a gradient of zero with respect toθ. Therefore, we can write the difference between the unconditional and conditional expectations as: ∇ θ E p t|0,T ,π 0,T ∥w θ (x,t)−w(x,t|x 0 ,x T )∥ 2 =∇ θ E p t ∥w θ (x,t)−w(x,t)∥ 2 ∇ θ E p t|0,T ,π 0,T [−2⟨w θ (x,t),w(x,t|x 0 ,x T )⟩] =∇ θ E p t [−2⟨w θ (x,t),w(x,t)⟩] ∇ θ E p t|0,T ,π 0,T [⟨w θ (x,t),w(x,t|x 0 ,x T )⟩] =∇ θ E p t [⟨w θ (x,t),w(x,t)⟩](6.36) Now, we aim to show that the expectations are equivalent. Starting from the ∇ θ E p t [⟨w θ (x,t),w(x,t)⟩] = Z R d ⟨w θ (x,t),w(x,t)⟩p t (x)dx = Z R d w θ (x,t), Z R d p t (x|x 0 ,x T ) p t (x) w(x,t|x 0 ,x T )π 0,T (x 0 ,x T )dx 0 dx T p t (x)dx (⋆) = Z R d Z R d ⟨w θ (x,t),w(x,t|x 0 ,x T )⟩ p t (x|x 0 ,x T ) p t (x) π T (x 0 ,x T )p t (x)dxdx 0 dx T = Z R d Z R d ⟨w θ (x,t),w(x,t|x 0 ,x T )⟩p t (x|x 0 ,x T )π 0,T (x 0 ,x T )dxdx 0 dx T = ∇ θ E p t|0,T ,π 0,T [⟨w θ (x,t),w(x,t|x 0 ,x T )⟩] (6.37) where (⋆) follows from factoring out the scalar values and applying Fubini’s theorem to change the order of integration, resulting in the equivalence between the marginal and conditional objectives.□ This result is well established in flow matching literature (Lipman et al., 2022; Tong et al., 2023a) as a way of training parameterized flows that approximate an intractable marginal distribution using empirical samples from the target data distribution. However, this marginal distribution does not yet solve the Schr ̈odinger bridge problem, as the coupled distributionπ 0,T from which (x 0 ,x T ) is sampled does not necessarily align with the entropic OT coupling. To establish how the conditional score and flow matching objective can be used to solve the (Dynamic SB Problem), we establish the following proposition. Proposition 6.4 (Score and Flow Matching Recovers the Schr ̈odinger Bridge (Proposition 3.4 in Tong et al. (2023b))). LetP ⋆ denote the path measure that solves the dynamic SB with marginal constraintsπ 0 ,π T ∈ P(R d ) and pure Brownian motion reference processσB. Consider the optimal endpoint lawπ ⋆ 0,T that solves the (Entropic OT Problem) with quadratic transport cost c(x,y) :=∥x−y∥ 2 : π ⋆ 0,T =arg min π 0,T ∈Π(π 0 ,π T ) Z R d ×R d ∥x−y∥ 2 dπ 0,T (x,y) + 2σ 2 t KL(π 0,T ∥π 0 ⊗ π T ) (6.38) 162 If the parameterized score functions ⋆ θ (x,t) and control driftv ⋆ θ (x,t) globally minimize the (Conditional Objective) under the coupling (x 0 ,x T )∼ π ⋆ 0,T , then the resulting stochastic process is given by the SDE: dX t = v ⋆ θ (X t ,t) + σ 2 t s ⋆ θ (X t ,t) dt + σ t dB t , X 0 ∼ π 0 (6.39) and generates the Schr ̈odinger bridge path measure P ⋆ . Proof Sketch. The key idea is that the (Conditional Objective) is minimized pointwise for all (x,t)∈ R d × [0,T ] when: v θ (x,t) = E (x 0 ,x T )∼p 0,T|t [u(x,t|x 0 ,x T )], s θ (x,t) = E (x 0 ,x T )∼p 0,T|t [∇ logp t (x|x 0 ,x T )](6.40) wherep 0,T|t is the posterior distribution over the endpoint lawπ 0,T given an intermediate statex which can be expressed using Bayes’ rule as: p 0,T|t (x 0 ,x T |x) = p t|0,T (x|x 0 ,x T )π 0,T (x 0 ,x T ) p t (x) (6.41) This minimizer is exactly the probability flow drift and score function of the mixture of Brownian bridges. From Section 4.1 Corollary 4.2, we showed ifπ ⋆ 0,T is chosen to be the entropic OT plan with quadratic transport cost, then this bridge mixture is precisely the Schr ̈odinger bridge, so the learned SDE recovers the Schr ̈odinger bridge P ⋆ .□ By leveraging a two-stage framework that first determines the optimal static SB coupling and learning the conditional velocity and score functions, score and flow matching provide a scalable, simulation- free framework for learning Schr ̈odinger bridges. However, a key limitation of this approach is that it requires explicit samples from both the source and target distributions to construct the conditional objectives and endpoint couplings. In many practical settings, such as those that are only known up to an unnormalized density or energy function, explicit samples may not be readily available. This motivates our discussion of alternative approaches that do not rely on paired or explicit samples from the target distribution. 6.5 Schr ̈odinger Bridge with Adjoint Matching Prerequisite: Section 3 In this section, we will explore how the adjoint matching framework (Domingo-Enrich et al., 2024) has been applied to efficiently solve the Schr ̈odinger bridge problem as described in Adjoint Schr ̈odinger Bridge Sampler (Liu et al., 2025). Adjoint matching (AM; Domingo-Enrich et al. (2024)) is a generative modeling framework that efficiently solves the stochastic optimal control (SOC) problem, which has been extended to various applications, including fine-tuning (Domingo-Enrich et al., 2024) and sampling (Havens et al., 2025; Park et al., 2025). The adjoint variable in the context of the Schr ̈odinger bridge problem refers to the gradient of the value functionV t (x) which defines the optimal control drift: u ⋆ (x,t) = σ t ∇ψ t (x) =−σ t ∇V t (x) =−σ t ∇J ⋆ (x,t;u)(6.42) Standard methods for solving for the adjoint variable by directly differentiating through the (SOC Objective) (Han et al., 2016) or directly match the target∇J ⋆ (x,t;u ⋆ ) with importance weighted matching objective (Domingo i Enrich et al., 2024; Rubinstein et al., 2004; Zhang et al., 2014), however, adjoint matching introduces a computationally favorable and fundamentally different approach. 163 Definition 6.5 (Adjoint State). The adjoint state, denoteda:C([t,T], R d )×[0,T]→ R d , of a stochastic optimal control (SOC) problem is defined by taking the gradient of the SOC objective defined in (SOC Objective) to get: a(X t:T ,t) =∇ Z T t 1 2 ∥u(X s ,s)∥ 2 + c(X s ,s) ds + Φ(X T ) ! (6.43) which yields the gradient field of J (x,t;u) in expectation: ∇J (x,t;u) = E X t:T ∼P u [a(X t:T ,t)|X t = x](6.44) The adjoint state can be solved backward in time, given the terminal conditiona(X t:T ,T;u) = ∇ x T Φ(X T ) by integrating: d dt a(X t:T ,t;u) =− a(X t:T ,t;u) ⊤ (∇(f (X t ,t) + σ t u(X t ,t)) +∇ c(X t ,t) + 1 2 ∥u(X t ,t)∥ 2 (6.45) which yields the alternative form: a(X t:T ,t;u) = Z T t ∇(f (X s ,s) ⊤ a(X t:T ,t;u) + σ t v(X s ,s)) +∇ c(X s ,s) + 1 2 ∥u(X s ,s)∥ 2 ds +∇Φ(X T )(6.46) Rather than directly matching the target adjoint vector field−σ t ∇J ⋆ (x,t;u ⋆ ), it considers an objective that matches the vector field generated by the current control−σ t ∇J(x,t;u), which bypasses the need for importance weighting while obtaining a optimizer gradient that is equal, in expectation to that of the target adjoint objective. Proposition 6.6 (Basic Adjoint Matching Yields the Optimal Control (Proposition 2 in Domingo-Enrich et al. (2024))). Consider basic adjoint matching objective defined with the adjoint state a : C([0,T ]; R d )× [0,T ]→ R d as: L basic-AM (u) := E X 0:T ∼P ̄u 1 2 Z T 0 u(X t ,t) + σ t ∇a(X t:T ,t; ̄ u) 2 dt , ̄ u = stopgrad(u)(6.47) where ̄ u=stopgrad(u) is the control drift where the gradient with respect touthat generates the path are not tracked, i.e., the pathX t:T cannot be differentiated through. Then, L basic-AM (u) has a unique minimizer that is equal to the optimal control u ⋆ . Proof. To derive the minimizer of the functional objective (6.47), we can compute the first variation ofL basic-AM by defining a slightly perturbed control drift (u+εv), wherev:R d ×[0,T]→ R d is an arbitrary vector field, to get: d dε L basic-AM (u + εv) = d dε E X 0:T ∼P ̄u " 1 2 Z T 0 ∥(u + εv)(X t ,t) + σ t ∇a(X t:T ,t; ̄ u)∥ 2 dt # ε=0 (6.48) 164 Applying the property d dε 1 2 ∥y + εz∥ 2 ε=0 = 1 2 [2(y + εz)z]| ε=0 =⟨y,z⟩, we have: d dε L basic-AM (u + εv) = E X t:T ∼P ̄u Z T 0 d dε 1 2 ∥u(X t ,t) + σ t a(X t:T ,t; ̄ u) + εv(X t ,t)∥ 2 dt ε=0 = E X t:T ∼P ̄u Z T 0 v(X t ,t),u(X t ,t) + σ t a(X t:T ,t; ̄ u) dt = E x∼p ̄u t Z T 0 v(x,t),u(x,t) + σ t E X t:T ∼P ̄u [a(X t:T ,t; ̄ u)|X t = x] |z must vanish point-wise at optimality dt (6.49) where we use the law of total expectation given that onlya(X t:T ,t; ̄ u) depends on the pathX t:T . Given thatv(X t ,t) is arbitrary, the only solution where the first variation (6.49) is zero is one where the expectation evaluates to zero for allv, which occurs if and only if for all (x,t), the following is satisfied: u(x,t) + σ t E X t:T ∼P ̄u [a(X t:T ,t; ̄ u)|X t = x] = 0(6.50) Therefore, we can write the functional derivative ofL basic-AM (u) with respect touevaluated pointwise at (x,t) as: δ δu L basic-AM (u)(x,t) = u(x,t) + σ t E X t:T ∼P ̄u [a(X t:T ,t; ̄ u)|X t = x](6.51) where the critical points satisfy: u(x,t) =−σ t E X t:T ∼P ̄u [a(X t:T ,t; ̄ u)|X t = x] (6.44) = −σ t ∇J (x,t;u)(6.52) To prove that anyuthat satisfies (6.52) for all (x,t)∈ R d ×[0,T] is the optimal controlu ⋆ , we establish the following Lemma. Proposition 6.7 (Fixed Point Solution to Optimal Control (Lemma 6 in Domingo-Enrich et al. (2024))). Consider a control driftu(x,t) that satisfiesu(x,t) =−σ t ∇J(x,t;u) for all (x,t)∈ R d ×[0,T]. Then, we have that the functionJ(·,·;u) :R d ×[0,T]→ Rsatisfies the Hamilton-Jacobi-Bellman equation defined in (2.72). Since the HJB equation has a unique solution, we can conclude that: ∀(x,t)∈ R d × [0,T ], J (x,t;u) = V t (x) =⇒ u(x,t)≡ u ⋆ (x,t) =−σ t ∇V t (x)(6.53) Proof. First, we decomposeJ(x,t;u) using (Bellman’s Principle of Optimality) which states that the optimal cost of timetis equal to the incremental cost over [t,t+ ∆t] and the cost of timet+ ∆t to get: J (x,t;u) = E Z t+∆t t + Z T t+∆t 1 2 ∥u(X s ,s)∥ 2 + c(X s ,s) ds + Φ(X T ) X t = x = E Z t+∆t t 1 2 ∥u(X s ,s)∥ 2 + c(X s ,s) ds X t = x + E [J (X t+∆t ,t + ∆t;u)|X t = x](6.54) SubtractingJ(x,t;u) from both sides, dividing by ∆t, and taking the continuous time limit ∆t→0, we get: 0 = lim ∆t E [J (X t+∆t ,t + ∆t;u)|X t = x]−J (x,t;u) ∆t + E R t+∆t t 1 2 ∥u(X s ,s)∥ 2 + c(X s ,s) ds X t = x ∆t 0 =A u J (x,t;u) + 1 2 ∥u(x,t)∥ 2 + c(x,t) 0 =A u J (x,t;u) + σ 2 t 2 ∥∇J (x,t;u)∥ 2 + c(x,t)(6.55) 165 Expanding the controlled generator A u J (x,t;u) with (Controlled Generator), we get: A u J (x,t;u) = ∂ t J (x,t;u) +⟨∇J (x,t;u),f (x,t) + σ t u(x,t)⟩ + σ 2 t 2 ∆J (x,t;u) = ∂ t J (x,t;u) +⟨∇J (x,t;u),f (x,t)⟩ +⟨∇J (x,t;u),σ t u(x,t)⟩ + σ 2 t 2 ∆J (x,t;u) = ∂ t J (x,t;u) +⟨∇J (x,t;u),f (x,t)⟩ +⟨∇J (x,t;u),−σ t σ t ∇J (x,t;u)⟩ + σ 2 t 2 ∆J (x,t;u) = ∂ t J (x,t;u) +⟨∇J (x,t;u),f (x,t)⟩−σ 2 t ∥∇J (x,t;u)∥ 2 + σ 2 t 2 ∆J (x,t;u)(6.56) Substituting this into (6.55) and completing the square, we get: 0 =A u J (x,t;u) + σ 2 t 2 ∥∇J (x,t;u)∥ 2 + c(x,t) 0 = ∂ t J (x,t;u) +⟨∇J (x,t;u),f (x,t)⟩−σ 2 t ∥∇J (x,t;u)∥ 2 + σ 2 t 2 ∆J (x,t;u) + σ 2 t 2 ∥∇J (x,t;u)∥ 2 + c(x,t) 0 = ∂ t J (x,t;u) +⟨∇J (x,t;u),f (x,t)⟩− σ 2 t 2 ∥∇J (x,t;u)∥ 2 + σ 2 t 2 ∆J (x,t;u) + c(x,t)(6.57) Rearranging terms, we recover the HJB equation from (Value HJB Equation): ∂ t J (x,t;u) =−⟨∇J (x,t;u),f (x,t)⟩− σ 2 t 2 ∆J (x,t;u) | z uncontrolled generator + σ 2 t 2 ∥∇J (x,t;u)∥ 2 − c(x,t) =A t J (x,t;u) + σ 2 t 2 ∥∇J (x,t;u)∥ 2 − c(x,t)(6.58) and since we defineJ(x,T;u) = Φ(x), we have shown thatJ(·,·;u) satisfies the HJB for all (x,t)∈ R d ×[0,T] givenu(x,t) =−σ t ∇J(x,t;u). By uniqueness of the solution to the HJB, we can conclude that J (x,t;u) = V t (x) and u(x,t) = u ⋆ (x,t) is the optimal control.□ The basic adjoint matching objectiveL basic-AM provides a theoretical foundation for the adjoint matching method but remains computationally inefficient as it requires differentiation through the cost functional, which depends on the full trajectory. Since we have shown thatu ⋆ (x,t) is the unique minimizer of (6.47), it can be written as the conditional expectation of the regression target: u ⋆ (x,t) = E X t:T ∼P u ⋆ [−σ t a(X t:T ,t;u ⋆ )|X t = x](6.59) At optimality, the forward control and adjoint state should be balanced everywhere, so we can multiply both sides of the equation by the Jacobian ∇u(x,t)∈ R d×d to get: u ⋆ (x,t) ⊤ ∇u ⋆ (x,t) = E X t:T ∼P u ⋆ −σ t a(X t:T ,t;u ⋆ ) ⊤ ∇u ⋆ (x,t)|X t = x (6.60) Since the left hand side depends only on x, we can rearrange to get: E X t:T [u ⋆ (x,t) ⊤ ∇u ⋆ (x,t) + σ t a(X t:T ,t;u ⋆ ) ⊤ ∇u ⋆ (x,t)|X t = x] = 0(6.61) which shows that at optimality, both terms that depend on the controlu(x,t) from the adjoint derivative (6.45) evaluate to zero. Leveraging this, Domingo-Enrich et al. (2024) introduces the lean adjoint state, which drops theu-dependent terms to obtain a computationally more efficient objective. 166 Definition 6.8 (Lean Adjoint State (Domingo-Enrich et al., 2024)). The lean adjoint state ̃ a:C([t,T], R d )×[0,T]→ R d is defined by the following differential equation which can be solved backward in time, given the terminal condition ̃ a(X,T ) =∇Φ(X T ) as: d dt ̃ a(X t:T ,t) =− ̃ a(X t:T ,t) ⊤ ∇f (X t ,t) +∇c(X t ,t) , ̃ a(X,T ) =∇Φ(X T )(6.62) Unlike the adjoint state defined in Definition 6.5, the lean adjoint state ̃ adoes not depend on the control u and does not require computing the Jacobian ∇u(x,t). ̃ a(X t:T ,t) = Z T t ∇f (X s ,s) ⊤ ̃ a(X s:T ,s) +∇f (X s ,s) ds +∇c(X T )(6.63) Using this lean adjoint state, we can construct an objective that directly matches the optimal control without requiring explicit computation of the value function or its gradients. Proposition 6.9 (Lean Adjoint Matching Yields the Optimal Control (Proposition 7 in (Domingo-Enrich et al., 2024))). Consider the lean adjoint matching objective defined with the lean adjoint state ̃ a : C([t,T ], R d )× [0,T ]→ R d as: L AM (u) := E X 0:T ∼P ̄u " 1 2 Z T 0 ∥u(X t ,t) + σ t ̃ a(X t:T ,t)∥ 2 dt # (Lean AM Objective) ̄ u := stopgrad(u) where ̄ u:=stopgrad(u) is the non-gradient tracking control drift. Then,L AM (u) has a unique minimizer which is exactly the optimal control u ⋆ . Proof. To prove this, we first establish the form of some critical point ˆ u ofL AM (u) and show that it is also a critical point of the basic adjoint matching lossL basic-AM . Then, we can apply the result from Proposition 6.7 to conclude that ˆ u is unique and is equal to the optimal control ˆ u = u ⋆ . Step 1: Derive the Critical Point. Since the form of the (Lean AM Objective) depends on a random variableX 0:T , to take the functional derivative, we need to evaluate it for some deterministic stateX t =x. Since ̃ a(X t:T ,t) is the only term inL AM that contains randomness inX t:T ∼ P ̄u after fixingX t =x, we add and subtract the conditional expectationE[ ̃ a(X t:T ,t)|X t ] to the expression inside the square in L AM to get: u(X t ,t) + σ t ̃ a(X t:T ,t) = u(X t ,t) + σ t E[ ̃ a(X t:T ,t)|X t ] + σ t ̃ a(X t:T ,t)− σ t E[ ̃ a(X t:T ,t)|X t ] | z vanishes when evaluated at a point X t =x (6.64) Then, substituting this expression back into the (Lean AM Objective), we have: L AM (u) = E X 0:T ∼P ̄u 1 2 Z T 0 ∥u(X t ,t) + σ t E[ ̃ a(X t:T ,t)|X t ] + σ t ̃ a(X t:T ,t)− σ t E[ ̃ a(X t:T ,t)|X t ]∥ 2 dt = E X 0:T ∼P ̄u 1 2 Z T 0 ∥u(X t ,t) + σ t E[ ̃ a(X t:T ,t)|X t ]∥ 2 dt + E X 0:T ∼P ̄u 1 2 Z T 0 ∥σ t ( ̃ a(X t:T )− E[ ̃ a(X t:T ,t)|X t ])∥ 2 dt |z not dependent on u Then, computing the functional derivative δ δu L AM (u)(x,t) evaluated at X t = x, we get: δ δu L AM (u)(x,t) = u(x,t) + σ t E[ ̃ a(X t:T ,t)|X t = x](6.65) 167 Since the first variation of δ δu L AM ( ˆ u ) of critical points ˆ u is zero, we get that the critical points of L AM ( ˆ u) satisfy: ∀x∈ R d , δ δu L AM ( ˆ u)(x,t) = 0 =⇒ ˆ u(x,t) =−σ t E[ ̃ a(X t:T ,t)|X t = x](6.66) which we will show is also a critical point for the basic adjoint matching objectiveL basic-AM in (6.47). Step 2: Matching Critical Point to Basic Adjoint Matching Objective. Using the same observation for obtaining the lean adjoint state, we note that at optimalty, the forward control and adjoint state can be balanced which allows us to multiply both sides of (6.66) with the Jacobian ∇ ˆ u(x,t) to get: ∇ ˆ u(X t ,t) ⊤ ˆ u(X t ,t) =−σ t ∇ ˆ u(X t ,t) ⊤ E[ ̃ a(X t:T ,t)|X t ] =⇒ E Z T t σ s ∇ ˆ u(X s ,s) ⊤ E[ ̃ a(X s:T ,s)|X s ] +∇ 1 2 ∥ ˆ u(X s ,s)∥ 2 ds X t = 0(6.67) Then, adding this zero-expectation balancing condition to the conditional expectation of the lean adjoint state E[ ̃ a(X t:T ,t)|X t ] using the definition in (6.63), we get: E[ ̃ a(X t:T ,t)|X t ] = E Z T t ∇f (X s ,s) ⊤ E[ ̃ a(X s:T ,s)|X s ] +∇f (X s ,s) ds +∇c(X T ) X t + E Z T t σ s ∇ ˆ u(X s ,s) ⊤ E[ ̃ a(X s:T ,s)|X s ] +∇ 1 2 ∥ ˆ u(X s ,s)∥ 2 ds X t | z zero-expectation balancing condition (6.67) = E Z T t ∇(f (X s ,s) + σ s ˆ u(X s ,s)) ⊤ E[ ̃ a(X s:T ,s)|X s ] +∇f (X s ,s) ds +∇ c(X T ) + 1 2 ∥ ˆ u(X s ,s)∥ 2 X t (6.68) Since the adjoint statea(X t:T ,t;u) also solves an equivalent integral equation from (6.46) with arbitrary v defined as: E[a(X t:T ,t;u)|X t ] = E Z T t ∇(f (X s ,s) + σ s v(X s ,s)) ⊤ E[ ̃ a(X s:T ,s)|X s ] +∇f (X s ,s) ds +∇ c(X T ) + 1 2 ∥v(X s ,s)∥ 2 X t (6.69) Settingv:= ˆ uin (6.69) yields the unique 27 solution atE[a(X t:T ,t; ˆ u)|X t ] =E[ ̃ a(X t:T ,t)|X t ] for allt∈[0,T]. Substituting this equality into the functional derivative of the basic adjoint matching loss L basic-AM (u) from (6.51), we get: δ δu L basic-AM ( ˆ u)(x,t) = ˆ u(x,t) + σ t E[a(X t:T ,t;u)|X t ] = ˆ u(x,t) + σ t E[ ̃ a(X t:T ,t)|X t ] (6.66) = −σ t E[ ̃ a(X t:T ,t)|X t ] + σ t E[ ̃ a(X t:T ,t)|X t ] = 0(6.70) which implies that all critical points ˆ uofL AM are critical points ofL basic-AM , and by Proposition 6.7, we conclude that ˆ u is unique and equal to the optimal control ˆ u = u ⋆ .□ 27 The uniqueness of the integral equation follows the expressing the difference between two arbitrary solutions as an integral inequality and observing that the largest squared distance between the two solutions is bounded by the integral of itself. By Gr ̈onwall’s inequality, this means their squared difference must equal zero, and thus the solution is unique. For full proof, see Proposition 8 in (Domingo-Enrich et al., 2024). 168 Given this general form of the adjoint matching objective, we can further simplify it for the case where the reference process is pure Brownian motion (f:= 0) and the running cost is zero (c:= 0), which results in the backward time evolution d dt ̃ adefined in (6.62) to vanish. Then, the lean adjoint state for allt∈[0,T] reduces to the gradient of the terminal cost with respect to the current state given by ̃ a(X t:T ,t) =∇ x T Φ(X T ). Therefore, the simplified adjoint matching objective becomes: L simple-AM (u) := E X 0:T ∼P ̄u " 1 2 Z T 0 ∥u(X t ,t) + σ t ∇ x T Φ(X T )∥ 2 dt # (Simplified AM Objective) ̄ u = stopgrad(u) To further simplify the objective, Havens et al. (2025) introduces the reciprocal adjoint matching objective, where rather than taking the expectation over full trajectoriesX 0:T ∼ P ̄u by repeatedly simulating the controlled SDE, we can leverage the key property that givenf:= 0,c:= 0, and X 0 = 0, the (Simplified AM Objective) depends only on (X t ,X T ) and the joint distribution of (X t ,X T ) under the optimal Schr ̈odinger bridge measure P ⋆ be factorized as: P ⋆ (X t ,X T ) = π T (X T )P ⋆ t|T (X t |X T ) = π T (X T )Q t|T (X t |X T )(6.71) sinceP ⋆ is in the reciprocal classR(Q), which shares the conditional bridgeP ⋆ t|T (X t |X T ) = Q t|T (X t |X T ) to the reference process. This gives us the objective: L RAM (u) := E X t ∼Q ̄u t|T ,X T ∼p ̄u T " 1 2 Z T 0 ∥u(X t ,t) + σ t ∇ x T Φ(X T )∥ 2 dt # (Reciprocal AM Loss) ̄ u = stopgrad(u) Crucially, the samplesX t ∼ Q t|T are independent conditioned on a terminal stateX T ∼ p ̄u T , enabling training on arbitrarily many intermediate samples given a single terminal stateX T . To minimize this objective, Havens et al. (2025) proposes an efficient iterative two-step algorithm called adjoint sampling. Adjoint Sampling Algorithm The adjoint sampling algorithm (Havens et al., 2025) is an iterative two step procedure: (i)First, simulate the forward-time controlled SDE using the non-gradient tracking control ̄ u to obtain a fixed set of samplesX T ∼ p ̄u T and evaluate the gradient of their terminal cost ∇ x T Φ(X T ). Then, store the set of samples in a replay bufferB=X i T ,∇ x i T g(X i T ) B i=1 . (i)Optimize the (Reciprocal AM Loss) by repeatedly sampling intermediate statesX t from the posterior X t ∼ Q t|T conditioned on samples from the replay buffer B. This process repeats until the optimal u ⋆ is reached. Intuitively, optimizing an arbitrary controlvwith (Reciprocal AM Loss) can be considered a reciprocal projection (Proposition 4.13), where we constrain the endpoint asp ̄u T (X T ) and a Markovian projection (Proposition 4.10) by projecting on the Markov bridge measureQ t|T (X t |X T ), where Q is the Markov reference Brownian motion. Proposition 6.10 (Convergence of Adjoint Sampling (Theorem 3.2 in Havens et al. (2025))). Consider optimizing the (Reciprocal AM Loss)L RAM given the current control ̄ uas projecting an arbitrary controlvonto the bridgeP ̄u (X 0:T ) =p ̄u T (X T )Q(X 0:T ) wherep ̄u T is the target 169 distribution: proj(u) = arg min v KL P v (X 0:T )∥p ̄u T (X T )Q(X 0:T ) (6.72) Then, at each iteration, we obtain the update: u n+1 = proj(u n )− δ δu L AM (proj(u n ))(6.73) where δ δu L AM is the functional derivative of the adjoint matching objectiveL AM defined in (6.65). The unique fixed pointu=proj(u) =u− δ δu L AM (u) is exactly the optimal control u ⋆ = proj(u ⋆ ). Proof. For this proof, we leverage the form of the critical point in (6.66) and the functional derivative in (6.65). Step 1: Unifying the Reciprocal Adjoint and Lean Adjoint Matching Objectives. First, we define a more general form of the reciprocal adjoint matching objective, where the matching target is an arbitrary vector field v, and the current control is u. L(u;v) = E X 0:T ∼P v " Z T 0 1 2 ∥u(X t ,t) + σ t ∇ x T Φ(X T )∥ 2 dt # (6.74) L RAM (u) :=L(u; proj( ̄ u)), L simple-AM (u) :=L(u; ̄ u)(6.75) which are equivalent definitions of the (Reciprocal AM Loss) and the (Simplified AM Objective) given ̄ u = stopgrad(u). Step 2: Derive Expression for Adjoint Sampling Iteration. Given the relationship between the lean adjoint and reciprocal adjoint matching losses defined in (6.75), we can apply (6.66) to write an optimal iteration of minimizing L(u; proj(u n )) as satisfying: u n+1 (x,t) =−σ t E P proj(u n ) [∇ x T Φ(X T )|X = x](6.76) Since theL RAM is justL AM with the target set toproj(u n ), we also apply (6.65) to write the functional derivative of L AM evaluated at proj(u n ) as: δ δu L AM (proj(u n )) = proj(u n ) + σ t E P proj(u n ) [∇ x T Φ(X T )|X = x] −σ t E P proj(u n ) [∇ x T Φ(X T )|X = x] |z =:u n+1 = proj(u n )− δ δu L AM (proj(u n )) u n+1 (x,t) = proj(u n )− δ δu L AM (proj(u n ))(6.77) which concludes the proof of (6.73) in the Proposition. Step 3: Fixed Point of Adjoint Sampling Iteration. We will now show thatuis a fixed point of adjoint sampling such thatu=proj(u) if and only ifuis a critical point of the (Lean AM Objective), which implies that u = u ⋆ by Proposition 6.9. First, we showu = proj(u) =⇒ u = u ⋆ . Supposeu=proj(u). By (6.73), we have δ δu L AM (proj(u)) = δ δu L AM (u) = 0. This means thatuis a critical point ofL AM and by Proposition 6.9, we have that u is unique and equal to the optimal control u = u ⋆ . 170 Figure 18: Schr ̈odinger Bridge with Adjoint Matching. Adjoint matching learns the Schr ̈odinger bridge by alternating between optimizing the forward half-bridge through the adjoint matching loss foru(x,t) and optimizing the backward half-bridge through the corrector matching loss for b Z T (X T ). Next, we showu = u ⋆ =⇒ u = proj(u). Supposeu=u ⋆ . Then, by Proposition 6.9, we knowu= u ⋆ . Since by definition,u ⋆ generates the optimal target of the projectionP ⋆ (X 0:T ) =π T (X T )Q(X 0:T ), projecting onto P ⋆ would yield itself, so u ⋆ = proj(u ⋆ ).□ Adjoint sampling (Box 6.5) provides a theoretically-grounded and computationally efficient method of obtaining the optimal controlu ⋆ when the posterior under the reference dynamics can be easily sampled as X t ∼ Q t|T (·|X T ), such as the linear Brownian motion case where f := 0 and the initial distribution is a dirac delta at zeroπ 0 :=δ 0 . However, its restriction to the dirac delta prior prevents more general settings with informative priors, such as Gaussians or task-specific priors. Additionally, adjoint matching requires the reference process to be memoryless, such that the joint distribution of (X 0 ,X T ) can be factorized asQ(X 0 ,X T ) =q 0 (X 0 )q T (X T ). This is to prevent the initial value function bias described in Box 3.1, where the optimal joint distributionP ⋆ (X 0 ,X T ) derived in Proposition 3.9 is dependent on an intractable initial value function V 0 (X 0 ): P ⋆ (X 0 ,X T ) = Q(X 0 ,X T )e −Φ(X T )+V 0 (X 0 ) (6.78) Although leveraging a memoryless reference drift guarantees sampling paths that generate the true target distributionπ T , it prevents the use of more informative prior distributions for sampling complex target distributions and crucially excludes all tasks where the goal is to map between distributions rather than simply sampling from a target distribution. This motivates using the (SB-SOC Objective) introduced in Section 3.2, which leverages the SB potentials (φ, ˆφ) defined by: u ⋆ (x,t) =−σ t ∇ logφ t (x), p ⋆ t (x) = φ t (x) ˆφ(x)(SB Optimality) where ( φ t (x) = R R d Q T|t (y|x)φ T (y)dy, π 0 (x) = φ 0 (x) ˆφ 0 (x) ˆφ t (x) = R R d Q t|0 (x|y) ˆφ 0 (y)dy, π T (x) = φ T (x) ˆφ T (x) Using these equations, we can replace the terminal cost inL simple-AM (Simplified AM Objective) with the definition in (SB Terminal Cost) given by∇Φ(x) =∇ log ˆφ T (x) π T (x) =∇ log ˆφ T (x)−∇ logπ T (x) to get the Schr ̈odinger bridge adjoint matching (SB-AM) loss. Definition 6.11 (Schr ̈odinger Bridge Adjoint Matching Objective). The Schr ̈odinger bridge adjoint matching (SB-AM) loss which solves the (SB-SOC Objective) is defined as: L SB-AM (u) := E X 0:T ∼P ̄u 1 2 Z T 0 ∥u(X t ,t) + σ t (∇ x T log ˆφ T (X T )−∇ x T logπ T (X T ))∥ 2 dt (SB-AM Objective) where ̄ u = stopgrad(u) is the non-gradient-tracking control drift. 171 To tractably compute∇ x T log ˆφ T (x T ), we define the bridge-matching or corrector matching objective (CM), which has been applied to both data-driven (Liu et al., 2023b; Shi et al., 2023) and sampling problems (Liu et al., 2025). Proposition 6.12 (Corrector Matching Objective). The gradient of the log Schr ̈odinger bridge potential∇ log ˆφ t (x) can be expressed as the minimizer of the Schr ̈odinger bridge corrector matching (SB-CM) loss defined as: L SB-CM ( b Z T ) := E p ⋆ 0,T h ∥ b Z T (X T )−∇ x T log Q T|0 (X T |X 0 )∥ 2 i (Corrector Matching Objective) where the minimizer defines the backward gradient of the log potential: ∇ log ˆφ T (x) = b Z ⋆ T = arg min b Z T E p ⋆ 0,T h ∥ b Z T (X T )−∇ x T log Q T|0 (X T |X 0 )∥ 2 i (6.79) Proof. Starting with the definition of ˆφ t from (SB Optimality), we have: ∇ log ˆφ t (x) = ∇ ˆφ t (x) ˆφ t (x) = 1 ˆφ t (x) ∇ Z R d Q t|0 (x|y) ˆφ 0 (y)dy = 1 ˆφ t (x) Z R d ∇Q t|0 (x|y) ˆφ 0 (y)dy = φ t (x) p ⋆ t (x) Z R d ∇ log Q t|0 (x|y)Q t|0 (x|y) ˆφ 0 (y)dy(6.80) Recalling from (SB-SOC Joint Density) that the joint density between any two timepointss≤ tis given byp ⋆ s,t (y,x) =Q(X s =y|X t =x)ˆφ s (y)φ t (x), we observe that we can replace the highlighted terms to get: ∇ log ˆφ t (x) = Z R d ∇ log Q t|0 (x|y) 1 p ⋆ t (x) Q t|0 (x|y) ˆφ 0 (y)φ t (x) |z =:p ⋆ 0,t (y,x) dy = Z R d ∇ log Q t|0 (x|y) p ⋆ 0,t (y,x) p ⋆ t (x) |z =p ⋆ 0|t (y|x) dy = Z R d ∇ log Q t|0 (x|y)p ⋆ 0|t (y|x)dy = E p ⋆ 0,t ∇ log Q t|0 (X t |X 0 )|X t = x (6.81) This can be rewritten as a regression objective where pairs (X 0 ,X t )∼ p ⋆ 0,t and we minimize a parameterized function b Z t (x) : R d × [0,T ]→ R d that minimizes the square loss: b Z ⋆ t := arg min h t E p ⋆ 0,t h ∥ b Z t (X t )−∇ log Q t|0 (X t |X 0 )∥ 2 i (6.82) where the minimizer aligns with the true log potential b Z ⋆ t (x) =∇ log ˆφ t (x). Fort=T, we obtain the expression ∇ log ˆφ T (x) that appears in L SB-AM (SB-AM Objective) as: b Z ⋆ T := arg min b Z T E p ⋆ 0,T h ∥ b Z T (X T )−∇ log Q T|0 (X T |X 0 )∥ 2 i (6.83) which is our definition for the (Corrector Matching Objective).□ Although this provides a concrete variational objective for obtaining∇ log ˆφ T (x) required to optimize (SB-AM Objective), it requires sampling (X 0 ,X T )∼ p ⋆ 0,T . In the case of sampling from an energy- based distribution, we have no explicit access to the target distributionp ⋆ T , which is exactly the challenge addressed by the adjoint Schr ̈odinger bridge sampler (Liu et al., 2025) algorithm. Rather than sampling pairs from the optimal joint distributionp ⋆ 0,T , we optimize the corrector b Z T using samples from the joint distributionp ̄u 0,T generated with the frozen control ̄ u . However, optimizinguusingL SB-AM (SB-AM Objective) also requires computing∇ x T log ˆφ T (X T ), which introduces cross-dependencies between the parameterized variables. 172 This naturally motivates an alternating optimization scheme, that switches between trainingu with the parameters of b Z T fixed and optimizing b Z T with the parameters ofufixed. Concretely, we can interpret this as optimizing a pair of forward and backward SDEs that are characterized by the control driftuin the forward time coordinatet∈[0,T] and correction term∇ log ˆφ t =∇ log ˆφ T−s in the backward time coordinate s = T − t∈ [0,T ], respectively: P u : dX t = (f (X t ,t) + σ t u(X t ,t))dt + σ t dB t ,X 0 ∼ π 0 (6.84) P b Z : d ̃ X s = h −f ( ̃ X s ,s) + σ 2 s ∇ log ˆφ T−s ( ̃ X s ) i ds + σ s dB s , ̃ X 0 ∼ π T (6.85) whereˆφ T−s is defined in the forward time coordinate via (SB Optimality) with the terminal constraint ∇ x T log ˆφ T (x T ) = b Z T (x T ). Then, optimizing both (SB-AM Objective) and (Corrector Matching Objective) reduces to determining the optimal pair (u ⋆ , b Z ⋆ T ), where the path measures generated by both the forward and backward SDEs align with the Schr ̈odinger bridge pathP u ⋆ =P b Z ⋆ =P ⋆ . Indeed, the alternating optimization scheme achieves this goal, which we will show by establishing that optimizing (SB-AM Objective) generates the optimal forward half-bridge (Proposition 6.13) and that optimizing (Corrector Matching Objective) generates the optimal backward half-bridge with from the terminal constraint (Proposition 6.14). Proposition 6.13 (Adjoint Matching Solves the Forward Schr ̈odinger Bridge (Theorem 4.1 in Liu et al. (2025))). Consider optimizing the (SB-AM Objective) with respect to the control driftu (k) at iterationk, then the path measure generated by the controlP u (k) solves the forward half bridge defined by the initial marginal constraint p 0 = π 0 given by: P u (k) = arg min P u n KL(P u ∥P b Z ); P u 0 = π 0 o (6.86) where P b Z is the path measure generated time-reversed SDE defined in (6.85). Proof. The goal of this proof is to show that optimizing the control driftu (k) using the corrector b Z (k−1) T from the previous iteration solves the forward half-bridge that is closest in KL divergence to the reverse-time dynamics generated from b Z (k−1) T . To do this, we first define the forward-time dynamics corresponding to the reverse-time SDE induced by b Z (k−1) T . Then, we apply Itˆo calculus to write the variational KL objective as an Itˆo integral which reduces the forward half-bridge matching objective to the (SB-AM Objective). Throughout the proof, we will denote the density induced by b Z (k−1) T asp b Z ≡ p b Z (k−1) and denoteˆz T as the potential which generates the gradient field b Z =∇ log ˆz T 28 . Step 1: Define the Forward SDE from the Backward Dynamics. Since the correctorh (k−1) defines the reverse-time dynamics via (6.85) following the time coordinates=T − t, we can define the corresponding forward-time dynamics for t using the (Time Reversal Formula) to get: P b Z : dX t = f (X t ,t)−σ 2 t ∇ log ˆφ t (X t ) + σ 2 t ∇ logp b Z t (X t ) | z (⋆) dt + σ t dB t , X T ∼ π T (6.87) The (Corrector Matching Objective) aims to matchP b Z with a path measure induced by some control drift u with the forward-time SDE: P u : dX t = (f (X t ,t) + σ t u(X t ,t) |z (⋄) )dt + σ t dB t , X 0 ∼ π 0 (6.88) 28 While the potentialˆz T is not the value typically learned in practice, it provides a useful notation for defining the terminal constraint that the dynamics induced by the learned gradient field b Z T 173 which minimizes the KL divergence toP b Z . Matching (⋄) to (⋆) can be expanded using the KL divergence derived via Girsanov’s theorem in Section 2.6 as: KL(P u ∥P b Z ) = E X u 0:T ∼P u Z T 0 1 2 u(X u t ,t) + σ t ∇ log ˆφ t (X u t )−∇ logp b Z t (X u t )) 2 dt (6.89) which depends on the integral of gradient terms∇ log ˆφ t (X t ) and∇ logp b Z t (X t ). Since we know that the Itˆo integral evaluates to the difference between boundary conditions, we can rewrite the objective using Itˆo’s calculus. Step 2: Apply Itˆo’s Caclulus to Simplify Integral. To expand the gradient terms, we can apply (Itˆo’s Formula) to the Itˆo processes defined bylog ˆφ t (X u t ) andlogp b Z t (X u t ) which contains the desired gradient term. Since the KL divergence (6.89) is respect to theP u , the underlying Itˆo process follows the SDE in (6.88), which we denote as (X u t ) t∈[0,T ] . First, for log ˆφ t (X u t ), we have: d log ˆφ t (X u t ) = ∂ t log ˆφ t (X u t ) + (f + σ t u)(X u t ,t) ⊤ ∇ log ˆφ t (X u t ) + σ 2 t 2 ∆ log ˆφ t (X u t ) dt + σ t ∇ log ˆφ t (X u t )dB t (6.90) To derive the expression for∂ t log ˆφ t (X u t ), we recall that it follows a deterministic integralˆφ t (x) = R R d Q t|0 (x|y)ˆφ 0 (y)dywith terminal conditionˆφ T =ˆz T . From Section 3.2, we show that any function applied to a Markov process defined by a terminal constraint satisfies the (Feynman-Kac Formula). Given the terminal constraint ˆφ T (x) = ˆz T (x), the (Feynman-Kac Formula) for ˆφ t is given by: ∂ t ˆφ t (x) =−∇· (f (x,t) ˆφ t (x)) + σ 2 t 2 ∆ ˆφ t (x), φ T (x) = ˆz T (x)(6.91) To obtain∂ t log ˆφ t from (6.91), we apply the chain rule and the divergence property∇·(f ˆφ t ) = (∇·f ) ˆφ t +f ⊤ ∇ ˆφ t to get: ∂ t log ˆφ t = 1 ˆφ t ∂ t ˆφ t = 1 ˆφ t −∇· (f ˆφ t ) + σ 2 t 2 ∆ ˆφ t =− 1 ˆφ t ((∇·f ) ˆφ t −f ⊤ ∇ ˆφ t ) + σ 2 t 2 ∆ ˆφ t ˆφ t =−∇·f −f ⊤ ∇ ˆφ t ˆφ t + σ 2 t 2 ∆ ˆφ t ˆφ t |z Laplacian trick =−∇·f −f ⊤ ∇ log ˆφ t + σ 2 t 2 (∆ log ˆφ t +∥∇ log ˆφ t ∥ 2 )(6.92) where the final equality is obtained from applying the Laplacian trick 29 . Finally, substituting the expression for ∂ t log ˆφ t from (6.92) into (6.90) and canceling like terms, we get: d log ˆφ t = −∇·f−f ⊤ ∇ log ˆφ t + σ 2 t 2 (∆ log ˆφ t +∥∇ log ˆφ t ∥ 2 ) + (f + σ t u) ⊤ ∇ log ˆφ t + σ 2 t 2 ∆ log ˆφ t dt + σ t ∇ log ˆφ t dB t = −∇·f + σ 2 t 2 ∥∇ log ˆφ t ∥ 2 + (σ t u) ⊤ ∇ log ˆφ t + σ 2 t ∆ log ˆφ t dt + σ t ∇ log ˆφ t dB t (log ˆφ-SDE) 29 derived as ∆ ˆφ t =∇·∇ ˆφ t =∇· ( ˆφ t ∇ log ˆφ t ) =∇ ˆφ t ·∇ log ˆφ t + ˆφ t ∆ log ˆφ t = ˆφ t ∇ ˆφ t ˆφ t ·∇ log ˆφ t + ∆ log ˆφ t = ˆφ t ∥∇ log ˆφ t ∥ 2 + ∆ log ˆφ t 174 Next, we apply Itˆo’s formula to ∂ t logp b Z t (X u t ): d logp b Z t (X u t ) = ∂ t logp b Z t (X u t ) + (f + σ t u)(X u t ,t) ⊤ ∇ logp b Z t (X u t ) + σ 2 t 2 ∆ logp b Z t (X u t ) dt + σ t ∇ logp b Z t (X u t )dB t (6.93) In this case, sincep b Z t (X u t ) is defined by the forward-time SDE (6.87), we can apply the (Fokker-Planck Equation) to write the time-evolution of the density ∂ t p b Z t (X u t ) as: ∂ t p b Z t =−∇· (f − σ 2 t ∇ log ˆφ t + σ 2 t ∇ logp b Z t ) |z drift of the SDE (6.87) p b Z t + σ 2 t 2 ∆p b Z t =−∇· ((f − σ 2 t ∇ log ˆφ t )p b Z t )−∇· (σ 2 t p b Z t ∇ logp b Z t |z =∇p b Z t ) + σ 2 t 2 ∆p b Z t =−∇· ((f − σ 2 t ∇ log ˆφ t )p b Z t )− σ 2 t ∇· (∇p b Z t ) | z ∆p b Z t + σ 2 t 2 ∆p b Z t =−∇· ((f − σ 2 t ∇ log ˆφ t )p b Z t )− σ 2 t 2 ∆p b Z t (6.94) Then, we can apply the chain rule to express ∂ t logp b Z t (x) from (6.94) as: ∂ t logp b Z t = 1 p b Z t ∂ t p b Z t = 1 p b Z t −∇· ((f − σ 2 t ∇ log ˆφ t )p b Z t )− σ 2 t 2 ∆p b Z t =− 1 p b Z t (p b Z t ∇· (f − σ 2 t ∇ log ˆφ t ) + (f − σ 2 t ∇ log ˆφ t ) ⊤ ∇p b Z t − σ 2 t 2 ∆p b Z t p b Z t =−∇· (f − σ 2 t ∇ log ˆφ t )− (f − σ 2 t ∇ log ˆφ t ) ⊤ ∇p b Z t p b Z t − σ 2 t 2 ∆p b Z t p b Z t | z Laplacian trick =−∇·f + σ 2 t ∆ log ˆφ t − (f − σ 2 t ∇ log ˆφ t ) ⊤ ∇ logp b Z t − σ 2 t 2 (∆ logp b Z t +∥∇ logp b Z t ∥ 2 )(6.95) Finally, substituting the expression for∂ t logp b Z t from (6.95) into (6.93) and canceling like terms, we obtain: d logp b Z t = −∇·f + σ 2 t ∆ log ˆφ t − (f − σ 2 t ∇ log ˆφ t ) ⊤ ∇ logp b Z t − σ 2 t 2 (∆ logp b Z t +∥∇ logp b Z t ∥ 2 ) + (f + σ t u) ⊤ ∇ logp b Z t + σ 2 t 2 ∆ logp b Z t dt + σ t ∇ logp b Z t dB t = −∇·f + σ 2 t ∆ log ˆφ t + σ 2 t ∇ log ˆφ ⊤ t ∇ logp b Z t − σ 2 t 2 ∥∇ logp h t ∥ 2 + (σ t u) ⊤ ∇ logp b Z t dt + σ t ∇ logp b Z t dB t (logp b Z t -SDE) 175 Observing that (log ˆφ-SDE) and (logp b Z t -SDE) have several matching terms, we can cancel them by subtracting (logp b Z t -SDE) from (log ˆφ-SDE) to get: d log ˆφ− d logp b Z t = −∇·f + σ 2 t 2 ∥∇ log ˆφ t ∥ 2 + (σ t u) ⊤ ∇ log ˆφ t + σ 2 t ∆ log ˆφ t dt + σ t ∇ log ˆφ t dB t − −∇·f + σ 2 t ∆ log ˆφ t + σ 2 t ∇ log ˆφ ⊤ t ∇ logp b Z t − σ 2 t 2 ∥∇ logp b Z t ∥ 2 + (σ t u) ⊤ ∇ logp b Z t dt + σ t ∇ logp b Z t dB t = (σ t u) ⊤ (∇ log ˆφ t −∇ logp b Z t ) + σ 2 t 2 ∥∇ log ˆφ t ∥ 2 + σ 2 t 2 ∥∇ logp b Z t ∥ 2 + σ 2 t ∇ log ˆφ ⊤ t ∇ logp b Z t dt + σ t ∇ log ˆφ t p b Z t dB t (6.96) where the terms inside the bracket are almost a perfect square ofu+∇ log ˆφ t −∇ logp b Z t . Completing the square with 1 2 ∥u∥ 2 , we get: d log ˆφ t − d logp b Z t = 1 2 u +∇ log ˆφ t −∇ logp b Z t 2 dt | z matches the KL divergence objective − 1 2 ∥u∥ 2 dt + σ t ∇ log ˆφ t p b Z t dB t 1 2 u +∇ log ˆφ t −∇ logp b Z t 2 dt = 1 2 ∥u∥ 2 dt + d log ˆφ t − d logp b Z t − σ t ∇ log ˆφ t p b Z t dB t (6.97) which recovers the integrand from our KL divergence objective. Step 3: Rewriting the KL Divergence.Using (6.97), we can rewrite the KL divergence objective in (6.89) as: KL(P u ∥P b Z ) = E X u 0:T ∼P u Z T 0 1 2 ∥u(X u t ,t)∥ 2 + d log ˆφ t (X u t )− d logp b Z t (X u t ) dt− Z T 0 σ t ∇ log ˆφ t (X u t ) p b Z t (X u t ) dB t |z =0 (Itˆo integral) = E X u 0:T ∼P u 1 2 ∥u(X u t ,t)∥ 2 dt + Z T 0 d log ˆφ t (X u t ) |z log ˆφ T −log ˆφ 0 − Z T 0 d logp b Z t (X u t ) | z logp b Z T −logp b Z 0 = E X u 0:T ∼P u 1 2 ∥u(X u t ,t)∥ 2 dt + log ˆφ T (X u T ) p b Z T (X u T ) − log ˆφ 0 (X u 0 ) p b Z 0 (X u 0 ) | z constant (6.98) where the term dependent onX u 0 is a constant with respect tousince we fix the initial distribution at X 0 ∼ π 0 . Therefore, substituting ˆφ T = ˆz (k−1) T into the objective, we have: KL(P u ∥P b Z ) = E X 0:T ∼P u 1 2 ∥u(X t ,t)∥ 2 dt + log ˆz (k−1) T (X T ) π T (X T ) + const (6.99) which is proportional to the (SB-AM Objective) up to an additive constant, and we conclude that the controlu (k) obtained from minimizing (SB-AM Objective) solves the forward half-bridge that minimizes KL(P u ∥P b Z ).□ However, we have already shown in (3.1) that solving the forward half-bridge for arbitrary prior distributions and reference drifts results in a mismatch of the target distribution, which motivated the definition of (Corrector Matching Objective) to define the terminal cost as Φ(x) :=log ˆφ T (x) π T (x) . While this definition of the terminal cost provably eliminates the initial value bias as shown in Section 3.2, we can further show that by optimizing (Corrector Matching Objective), we obtain the optimal corrector b Z (k) T that induces the backward half-bridge which is closest in KL to the forward dynamics 176 defined by any arbitrary controluwhile correcting for the bias created at the terminal distribution π T . Proposition 6.14 (Corrector Matching Solves the Backward Schr ̈odinger Bridge (Theorem 4.2 in Liu et al. (2025))). Consider optimizing the (Corrector Matching Objective) with the control driftu (k) from thekth iteration, then the path measure generated by the corrector b Z (k) T solves the backward half bridge defined by the terminal constraint p T = π T as: P b Z = arg min P b Z n KL(P u (k) ∥P) : p b Z T = π T o (6.100) whereP u (k) is the path measure generated from the learned control at iterationkdefined in (6.84). Proof. This proof starts by defining the time-reversal of the forward half-bridge generated byu (k) from thekth iteration of the algorithm and showing that optimizing (Corrector Matching Objective) yields the optimal reverse-time dynamics that enforce the terminal constraint. Step 1: Time Reversal of Forward Controlled SDE. Given the control drift from the current iteration of the optimization algorithmu (k) , we can define the corresponding backward SDE following the reversed time coordinate s := T − t using the (Time Reversal Formula) to get: P u (k) : d ̃ X s = −f ( ̃ X s ,s)− σ s u (k) ( ̃ X s ,s) + σ 2 s ∇ logp u (k) s ( ̃ X s ) ds + σ s dB s , ̃ X 0 ∼ π T (6.101) where the highlighted terms are defined by the controlu (k) . Since we have shown thatu (k) solves the forward half-bridge in Proposition 6.13, it satisfies the SB equations given by: u (k) t (x,t) = σ t ∇ logφ t (x) ( φ t (x) = R R d Q T|t (y|x)φ T (y)dy, π 0 (x) = φ 0 (x) ˆφ 0 (x) ˆφ t (x) = R R d Q t|0 (x|y) ˆφ 0 (y)dy, p u (k) T (x) = φ T (x) ˆφ T (x) (6.102) which does not necessarily satisfy the terminal constraintp ⋆ T =π T . The goal of the (Corrector Matching Objective) is to generate the backward half-bridge that minimizes the divergence from the forward half-bridge while constraining the terminal marginal to p b Z T = π T . Step 2: Deriving the Matching Objective. To do this, we aim to match some arbitrary control v(X s ,s) to the time-reversal of the forward half-bridge in (6.101) where the initial states ̃ X 0 ∼ π T are sampled from the target marginalπ T . To define the matching loss as a KL divergence, letv(x,s) be an arbitrary control drift defining the reverse-time SDE initialized at π T : P v :d ̃ X s = (−f ( ̃ X s ,s) + σ s v( ̃ X s ,s))ds + σ s dB s , ̃ X 0 ∼ π T (6.103) P u (k) :d ̃ X s = −f ( ̃ X s ,s) + σ s (−σ s ∇ logφ s ( ̃ X s ) + σ s ∇ logp u (k) s ( ̃ X s )) ds + σ s dB s , ̃ X 0 ∼ p u (k) T (6.104) which we will show matches the result obtained from minimizing (Corrector Matching Objective) at optimality. Now, expanding the KL divergence KL(P u (k) ∥P v ) as shown in Section 2.6, we have: KL(P u (k) ∥P v ) = E X 0:T ∼P u (k) Z T 0 1 2 (−σ s ∇ logφ s ( ̃ X s ) + σ s ∇ logp u (k) s ( ̃ X s ))−v( ̃ X s ,s) 2 ds (6.105) Minimizing (6.105) yields for all (x,s), the following expression for v(x,s): v ⋆ (x,s) =−σ s ∇ logφ T−s (x) + σ s ∇ logp u (k) s (x) = σ s ∇ log φ s (x) p u (k) s (x) | z = ˆφ T−s (x) = σ s ∇ log ˆφ T−s (x)(6.106) 177 where we use the (SB Optimality) which definesp u (k) t (x) =φ t (x)ˆφ t (x). Substituting this expression into the backward path measureP v in (6.103), we have the optimal half-bridge is generated by the SDE: d ̃ X s = (−f ( ̃ X s ,s) + σ 2 s ∇ log ˆφ T−s ( ̃ X s ))ds + σ s dB s ,ˆφ T = p u (k) T φ T (6.107) which is fully characterized by the terminal conditionˆφ T = p u (k) T φ T from whichˆφ T−s can be defined using (SB Optimality). From Proposition 6.12, we show that optimizing (Corrector Matching Objective) yields b Z (k) T =∇ log ˆφ T (x) given the optimal SB densityp ⋆ t , so applying the same logic, we have: b Z (k) T := arg min b Z T E p u (k) 0,T b Z T (X T )−∇ log Q T|0 (X T |X 0 ) 2 (6.12) = ∇ log ˆφ T (6.108) which yields the same backward time SDE through the terminal constraint as the optimal driftv ⋆ , and we have shown that optimizing (Corrector Matching Objective) is equivalent to finding the optimal reverse-time dynamics that correct the SB forward-time SDE such that it satisfies the terminal constraint.□ The results from Proposition 6.13 and Proposition 6.14 indicate that alternating between optimizing (SB-AM Objective) and (Corrector Matching Objective) is equivalent to alternating between solving the forward Schr ̈odinger half-bridge that satisfies the initial marginal to solving the backward Schr ̈odinger half-bridge that satisfies the target marginal. This alternating scheme is reminiscent of our discussion of Sinkhorn’s algorithm from Section 1.5, but now adapted using an efficient matching objective. Just like the adjoint matching algorithm (Box 6.5), we can optimize (SB-AM Objective) and (Corrector Matching Objective) by repeatedly optimizing over samples from a replay buffer. Remark 6.15 (Adjoint and Corrector Matching Doesn’t Require Target Samples). We highlight that optimizing (SB-AM Objective) and (Corrector Matching Objective) does not require explicit samples fromX T ∼ π T since the samples used to compute the objective are purely from sampling (X 0 ,X T )∼ p u 0,T from the SDE induced byu. The dependence onπ T only appears when computing the loss, which can be computed as a probability underπ T , which can be empirical or a pre-defined potential energy function, as we will discuss further in Section 8.3. Throughout this section, we have demonstrated how the adjoint state can be used as an efficient variational framework for solving Schr ̈odinger bridge problems by alternating between two tractable objectives that correspond to the forward and backward Schr ̈odinger half-bridges. Rather than directly optimizing over path measures, the method reduces the problem to learning the forward control drift and the backward correction through cheap matching objectives that can be evaluated using trajectories generated by the current dynamics. This formulation yields several practical advantages: (i)It avoids the need for expensive likelihood ratios or full path-space KL computations, replacing them with local drift-matching losses that are straightforward to estimate. (i) The alternating optimization naturally mirrors the structure of Sinkhorn iterations in entropic optimal transport, providing convergence guarantees to the optimal SB control u ⋆ . (i) The corrector matching objective does not require explicit samples from the target distribution, enabling training on unknown energy-based target densities. 178 (iv)Since the objectives do not require backpropagation through SDE trajectories or maintaining the full trajectory in memory, it can easily scale to high-dimensional systems. These properties make adjoint matching a practical and flexible approach for learning Schr ̈odinger bridges in complex generative modeling settings. 6.6 Closing Remarks for Section 6 In this section, we explored several generative modeling frameworks grounded in Schr ̈odinger bridge (SB) theory. We began with the classical score-based generative modeling paradigm (Section 6.1), which can be interpreted as a specialized instance of the SB formulation in which the forward process is fixed, and learning focuses on estimating the reverse-time dynamics. Building on this perspective, we then leveraged SB theory to generalize to controlled forward processes, where both the forward and backward control drifts can be learned through likelihood-based training (Section 6.2). Next, we introduced an alternative viewpoint through the Iterative Markovian Fitting (IMF) procedure and the diffusion Schr ̈odinger bridge matching algorithm (Section 6.3), which draws on the Markov and reciprocal projection theory developed in Section 4.5. In this formulation, solving the Schr ̈odinger bridge problem can be understood as performing iterative projections in path space. To overcome the limitations in optimizing over full stochastic trajectories in the path space from the previous approaches, we conclude with two approaches for learning Schr ̈odinger bridges with efficient matching objectives, including score and flow matching (Section 6.4) and adjoint matching (Section 6.5), which locally optimize the control drift to generate trajectories consistent with the optimal Schr ̈odinger bridge dynamics. Overall, this section builds the intuition behind the core generative modeling frameworks that leverage Schr ̈odinger bridge theory. While it is not intended to be an exhaustive review of algorithmic developments in the field, it should provide the core theoretical foundations needed to understand a broad class of modern generative modeling techniques. So far, we have restricted our attention to the continuous state space, where data is represented as continuous-valued vectors inR d . The structure of the continuous state space is required for many ideas developed throughout this guide, including stochastic differential equations, path measures, and Brownian motion. This naturally raises the question: How does Schr ̈odinger bridge theory extend to the discrete state space, where states belong to a finite or countable set rather than a continuous vector space? This is precisely the question that we explore in the next section: (i)We will introduce the concept of stochastic processes in the discrete state space as continuous- time Markov chains (CTMCs), where the control drift in SDEs takes the analogous form of a transition rate matrix in discrete state spaces. (i)While the structure of the discrete Schr ̈odinger bridge problem remains the same, we will introduce fundamental differences in the KL divergence in discrete state spaces. This will provide the theoretical grounding for our discussion on solving the discrete Schr ̈odinger bridge problem with generative modeling, which adapts several ideas developed in the previous sections for the discrete state space. In doing so, we extend the Schr ̈odinger bridge framework from diffusion processes on continuous spaces to jump processes on discrete state spaces, laying the groundwork for an even broader class of generative modeling methods. 179 7. From Continuous to Discrete State Space Now that we have built the foundation required to understand and construct the Schr ̈odinger bridge where the states exist as continuous latent vectors in some state spaceX ⊆ R d , we will now take a detour to the discrete state space, where states exist as probabilities of existing in a finite set of discrete states. We will see that rather than representing dynamics with velocity fields that transport states via smooth lines, dynamics in the discrete state spcae are represented with transition rates that characterize the instantaneous change in the probabilities of existing in each state. In this section, we introduce discrete state path measures not as SDEs but as continuous-time Markov chains (CTMCs) defined by their rate matrices (Section 7.1). Leveraging the theory of CTMCs, we define the discrete Schr ̈odinger bridge problem, and extend the definitions for the Radon-Nikodym derivative (RND) and KL divergence to CTMCs (Section 7.2). Then, we analyze two methods of solving the discrete SB problem which mirror the continuous state space, starting with the stochastic optimal control formulation (Section 7.3 and 3.2) and concluding with the discrete analog of Iterative Markovian Fitting using Markovian and reciprocal projections (Sections 7.5 and 7.6). 7.1 Continuous-Time Markov Chains The discrete state space can be defined as a finite set of statesX=1,...,dand the probability simplex ∆ d−1 over the d discrete states given by: ∆ d−1 = ( x = (x 1 ,...,x d )∈ R d x i ∈ [0, 1], d X i=1 x i = 1 ) (7.1) To define a stochastic path measurePthat lies on the simplex ∆ d−1 , we introduce the theory of continuous-time Markov chains (CTMCs) in the probability space (Ω, Pr), where Ω∈ D([0,T],X) is the space of left-limited and right-continuous (c ́adl ́ag) 30 paths overXandPris the probability measure over events. A CTMC is a stochastic process that evolves over timeX 0:T whose probability law is defined by a time-dependent generator or transition rate matrix (Q t ∈ R X×X ) t∈[0,T ] of the form: Q t (x,y) = lim ∆t→0 1 ∆t (Pr(X t+∆t = y|X t = x)− 1 x=y )(7.2) which defines the instantaneous rate of transitioning from statex ∈ Xto statey ∈ Xat timet. Since the state transitions must remain on the probability simplex, the transition rates satisfy the following conditions: ∀x̸= y, Q t (x,y)≥ 0, X y∈X Q t (x,y) = 0(7.3) A generatorQuniquely defines a path measureP∈P(Ω), under which we can define the transition probability over a discrete time interval [t,t + ∆t] as: P(X t+∆t = y|X t = x) = ( ∆tQ t (x,y) +O(∆t 2 )y ̸= x 1− ∆t P z̸=x Q t (x,z) +O(∆t 2 ) y = x (7.4) which can be derived from the definition of the generator as the instantaneous jump rate, and multiplying that by the time interval ∆tfor all statesy ̸=xwithO(∆t 2 ) accounting for the error of multiple jumps within the discrete interval. To ensure that the probabilities are normalized and sum to one, i.e., P y∈X P (X t+∆t =y|X t =x) = 1, we subtract all probabilities of leaving the statexas to get the probability of remaining at x. 30 piecewise continuous paths with jump discontinuities where the right limit is the actual value. 180 Figure 19: Piecewise C`adl`ag Path of Continuous-Time Markov Chains. Illustration of a c`adl`ag stochastic trajectory, which is right-continuous with left limits. Between jump timest 1 ,...,t 5 the process evolves continuously, while discontinuities occur at the marked times. The filled markers indicate the value of the processX t (the right limit), while the open markers denote the left limit X t − . Given that CTMCs are c ́adl ́ag paths, we denote the left limit of a stateX t asX t − =lim s↑t X s , whereX t − ̸ =X t at jump times from stateX t − to stateX t . A key property of CTMCs is that they satisfy the Kolmogorov forward equation which define the time evolution of the path. Lemma 7.1 (Kolmogorov Forward Equation for CTMCs). The forward-time dynamics of a CTMCX 0:T with probability measurep t (·) :=Pr(X t =·) and generatorQ t satisfies the Kolmogorov forward equation defined as: ∀x∈X, ∂ t p t (x) = X y∈X Q t (x,y)p t (y) = X x̸=y (Q t (y,x)p t (y)−Q t (x,y)p t (x))(7.5) where that the probability measurep t (·) is unique given a pair of endpoint conditions at t∈0,T and t7→ Q t is continuous over time t∈ [0,T ]. Proof. This proposition can be shown simply by defining a forward transition probability over the discrete time increment [t,t+ ∆t] and taking a limit as ∆t→0 to get the expression for∂ t p t . First, using (7.4) we have: p t+∆t (x) = X y∈X Pr(X t+∆t = x|X t = y)p t (y) (7.4) = X y∈X (1 x=y + ∆tQ t (x,y) +O(∆t 2 ))p t (y) = p t (x) + ∆t X y∈X Q t (y,x)p t (y) +O(∆t 2 )(7.6) 181 Figure 20: Continuous-Time Markov Chains (CTMC). A CTMC defines a stochastic process on a finite state spaceX=1,...dthat evolves through jumps between discrete states governed by a time-dependent transition rate matrixQ t . At each jump timet k , the system transitions from the current stateX t k to a different stateyaccording to the rateQ t (X t k ,y). The evolution of the probability distributionp t (x) satisfies the Kolmogorov forward equation, which balances incoming and outgoing probability flows induced by the transition rates. Taking the continuous time limit as ∆t→ 0, we have: ∂ t p t (x) = lim ∆t→0 " ∆t X y∈X Q t (y,x)p t (y) +O(∆t 2 ) # = X y∈X Q t (y,x)p t (y) = Q t (x,x)p t (x) + X y̸=x Q t (y,x)p t (y) =− X y̸=x Q t (y,x)p t (y) + X y̸=x Q t (y,x)p t (y) = X x̸=y (Q t (y,x)p t (y)−Q t (x,y)p t (x))(7.7) which is exactly the Kolmogorov forward equation defined in (7.5). To prove uniqueness, we can write (7.7) in vector form as: ∀t∈ [0,T ] : −∂ t p t = Q t p t ,s.t. p t = (p t (x) : x∈X )∈ R |X| (7.8) which is a linear ODE in R |X| . Since t7→ Q t is continuous, linear ODEs have a unique solution. □ Having derived the Kolmogorov forward equation, which describes how the state distributionp t evolves forward under the time-dependent generatorQ t , we now ask how CTMC dynamics evolve backwards from a terminal constraint. As we have seen in the continuous state space, the idea of terminal conditioning is the foundation for solving the Schr ̈odinger bridge problem, as it allows us to define the optimal dynamics that generate a target distribution. The idea of evolving CTMC dynamics backward is exactly captured by the Kolmogorov backward equation, which we define next. Lemma 7.2 (Kolmogorov Backward Equation for CTMCs). Consider a CTMCX 0:T under the measurePwith generatorQ t and letφ t (x) :=E[φ(X T )|X t =x] given an arbitrary test functionφ:X → R. Then, the reverse-time dynamics ofφ t satisfies the Kolmogorov backward equation defined as: ∀x∈X, −∂ t φ(x) = X y∈X φ t (y)Q t (x,y) = X y̸=x (φ(y)− φ(x))Q t (x,y), φ T (x) = φ(x)(7.9) 182 which admits a unique solution φ when t7→ Q t is continuous over all t∈ [0,T ]. Proof. Sinceφ t (x) can be considered some cost function to go fromX t =xto the terminal state X T , we can expand the inner functionφ(X T ) using the law of total expectation to writeφ t (x) with respect to a discrete time step φ t+∆t (x): φ t (x) = E[φ(X T )|X t = x] = E[E[φ(X T )|X t+∆t ]|X t = x] = E[φ t+∆t (X t+∆t )|X t = x](7.10) Then, applying the transition probability of a discrete time step defined in (7.4), we have: φ t (x) = E[φ t+∆t (X t+∆t )|X t = x] = X y∈X φ(y)P(X t+∆t = y|X t = x) = X y∈X φ t+∆t (y) 1 x=y + ∆tQ t (x,y) +O(∆t 2 ) = φ t+∆t (x) + ∆t X y∈X φ t+∆t (y)Q t (x,y) +O(∆t 2 )(7.11) Now, subtractingφ t+∆t (x) from both sides, dividing by ∆t, and taking the continuous time limit ∆t→ 0, we get the reverse time dynamics −∂ t φ t (x) as: −∂ t φ t (x) = lim ∆t→0 φ t (x)− φ t+∆t (x) ∆t = X y∈X φ t (y)Q t (x,y) = X y̸=x φ t (y)Q t (x,y) + φ t (x)Q t (x,x) = X y̸=x φ t (y)Q t (x,y)− X y̸=x φ t (x)Q t (x,y) = X y̸=x (φ t (y)− φ t (x))Q t (x,y)(7.12) which is the Kolmogorov backward equation. To prove uniqueness, we follow the proof of the forward equation and write (7.12) in vector form as−∂ t φ t =Q t φ t , which is a linear ODE inR |X| . Since t7→ Q t is continuous, linear ODEs have a unique solution.□ CTMCs can be interpreted as the discrete analog of stochastic differential equations (SDEs) in the continuous state space, where the time-dependent generator conditioned on generatorQ t (x,·) : X×[0,T]→ R d is analogous to the time-dependent driftu(x,t) of an SDE. Since we have characterized the forward and reverse evolution of path-measure CTMCs, we are ready to formulate the discrete- state-space analog of the Schr ̈odinger bridge problem, which aims to recover a CTMC that satisfies a pair of marginal distributions while remaining close to a reference CTMC. 7.2 Discrete Schr ̈odinger Bridge Problem In this section, we formulate the discrete Schr ̈odinger bridge problem for continuous-time Markov chains (CTMCs) on a finite state spaceX=1,...,d. Just like the continuous-space Schr ̈odinger bridge problem, the objective is to find a path measureP ⋆ that is closest, in relative entropy or KL divergence, to a given reference processQ, while matching prescribed marginal constraints p 0 = π 0 and p T = π T . Since the control drift in continuous SB theory is denotedu(x,t) :R d ×[0,T]→ R d , we useQ u t (x,·) : X×[0,T]→ R d×d to denote the CTMC generator of the controlled path measureP u ∈P(C([0,T];X)), which we aim to optimize to recover the Schr ̈odinger bridge. To denote the generator of the reference process Q∈P(C([0,T ];X )), we use Q 0 t (x,·) :X × [0,T ]→ R d×d . Definition 7.3 (Discrete Schr ̈odinger Bridge Problem). Consider a reference CTMC measureQ with generatorQ 0 t and a pair of marginal distributions in the discrete state spaceπ 0 ,π T ∈P(X). The discrete Schr ̈odinger bridge problem aims to determine the optimal CTMC path measureP ⋆ with generatorQ ⋆ t wherep 0 =π 0 andp T =π T that solves the minimization 183 problem: P ⋆ =arg min P u ∈P(C([0,T ];X ) KL(P u ∥Q) : p 0 = π 0 ,p T = π T (Discrete SB Problem) where the KL divergence KL(·∥·) between CTMCs is defined as: KL(P u ∥Q) = E X 0:T ∼P u Z T 0 X y̸=X t Q u t log Q u t Q 0 t +Q 0 t −Q u t (X t ,y) dt(7.13) Solving the discrete SB problem yields the CTMC path measure that minimizes the discrepancy from the reference jump process, which penalizes differences in jump intensities, jump times, and transition structure over the time interval t∈ [0,T ]. To define the form of the KL divergence, we first derive the Radon-Nikodym derivative between CTMC path measures, which defines the probability ratio of a CTMCX 0:T under two path measures P and P ′ with generators Q t and Q ′ t . Proposition 7.4 (Radon-Nikodym Derivative Between CTMCs). Consider two CTMC path measuresPandP ′ with generatorsQandQ ′ t and initial distributionsp 0 =π 0 andP ′ 0 =π ′ 0 . Assume that theπ ′ 0 ≪ π 0 andP ′ ≪ P, where the generators satisfyQ(x,y) = 0 =⇒ Q(x,y) = 0. Then, the logarithm of the Radon-Nikodym derivative is given by: log dP ′ dP (X 0:T ) = log dπ ′ 0 dπ 0 (X 0 ) + X t:X t − ̸=X t log Q ′ t (X t − ,X t ) Q t (X t − ,X t ) + Z T 0 X y̸=X t (Q t −Q ′ t )(X t ,y)dt (7.14) Proof. To derive the RND for CTMCs, we first consider the discrete-time case, where we break down the time horizon into time steps 0 =t 0 < t 1 <· < t k <· < t K−1 < t K =Twhere the time intervals are separated by ∆t. Then, we can write the log ratio of a discrete path (X t k ) k∈0,...,K under the probability measures P and P ′ as: log dP ′ dP (X 0:T ) = log dπ ′ 0 dπ 0 (X 0 ) + K−1 X k=0 log dP ′ (X t k+1 |X t k ) dP(X t k+1 |X t k ) +O(∆t)(7.15) From (7.4), we can derive the probability ratio for two distinct cases given an interval [t k ,t k+1 ]: the case of at least one change in state in [t k ,t k+1 ] and the case of no change in state in [t k ,t k+1 ]. First, for the case when at least one jump is made, the log ratio becomes: log P ′ (X t k+1 |X t k ) P(X t k+1 |X t k ) = log ∆tQ ′ t k (X t k ,X t k+1 ) +O(∆t 2 ) ∆tQ t k (X t k ,X t k+1 ) +O(∆t 2 ) = log Q ′ t k (X t k ,X t k+1 ) Q t k (X t k ,X t k+1 ) +O(∆t 2 )(7.16) For the case when no jumps are made, the log ratio becomes: log P ′ (X t k+1 |X t k ) P(X t k+1 |X t k ) = log 1− ∆t P y̸=X t k Q ′ t k (X t k ,y) +O(∆t 2 ) 1− ∆t P y̸=X t k Q t k (X t k ,y) +O(∆t 2 ) = ∆t X y̸=X t k Q t k (X t k ,y)−Q ′ t k (X t k ,y) +O(∆t 2 )(7.17) 184 Substituting (7.16) and (7.17) into (7.15) and taking the continuous time limit ∆t→0 andK →∞, we have: log dP ′ dP (X 0:T ) = lim ∆t→0 log dπ ′ 0 dπ 0 (X 0 ) + K−1 X k=0 log Q ′ t k (X t k ,X t k+1 ) Q t k (X t k ,X t k+1 ) + K−1 X k=0 ∆t X y̸=X t k Q t k (X t k ,y)−Q ′ t k (X t k ,y) +O(∆t) = log dπ ′ 0 dπ 0 (X 0 ) + X t:X t − ̸=X t log Q ′ t (X t − ,X t ) Q t (X t − ,X t ) + Z T 0 X y̸=X t (Q t −Q ′ t )(X t ,y)dt(7.18) which is the exact form of the log RND between P ′ and P.□ Using this result, we can easily derive the KL divergence between CTMC path measures, which is the foundation of the discrete Schr ̈odinger bridge objective. Corollary 7.5 (KL Divergence Between CTMCs). The KL divergence between two CTMC path measures P and P ′ with generators Q and Q ′ t is defined as: KL(P ′ ∥P) = KL(π ′ 0 ∥π 0 ) + E X 0:T ∼P ′ Z T 0 X y̸=X t Q ′ t log Q ′ t Q t +Q t −Q ′ t (X t ,y) dt(7.19) and can be equivalently written as: KL(P ′ ∥P) = KL(π ′ 0 ∥π 0 ) + E X 0:T ∼P ′ Z T 0 X y̸=X t Q ′ t log Q ′ t Q t (X t ,y) + (Q ′ t −Q t )(X t ,X t ) dt (7.20) Proof. Recalling that the KL divergenceKL(P ′ ∥P) is simply the expectation of the log RND over paths from the first argument, we have: KL(P ′ ∥P) = E X 0:T ∼P ′ log dP ′ dP = E X 0:T ∼P ′ log dπ ′ 0 dπ 0 (X 0 ) + X t:X t − ̸=X t log Q ′ t (X t − ,X t ) Q t (X t − ,X t ) + Z T 0 X y̸=X t (Q t −Q ′ t )(X t ,y)dt = E X 0:T ∼P ′ log dπ ′ 0 dπ 0 (X 0 ) | z (⋆) + E X 0:T ∼P ′ X t:X t − ̸=X t log Q ′ t (X t − ,X t ) Q t (X t − ,X t ) | z (♦) + E X 0:T ∼P ′ Z T 0 X y̸=X t (Q t −Q ′ t )(X t ,y)dt | z (⋄) (7.21) The first term (⋆) can be written as: E X 0:T ∼P ′ log dπ ′ 0 dπ 0 (X 0 ) |z only depends on X 0 = E X 0 ∼π ′ 0 log dπ ′ 0 dπ 0 (X 0 ) = KL(π ′ 0 ∥π 0 )(⋆) 185 To write the second term (♦) as an integral, we can consider the discrete time case for 0 =t 0 <· < t k <· < t K = T , distribute the expectation, and take the continuous time limit as follows: E (X t k ) k∈0,...,K ∼P ′ K−1 X k=0 1[X t k+1 ̸= X t k ] log Q ′ t k (X t k ,X t k+1 ) Q t k (X t k ,X t k+1 ) = K−1 X k=0 E P ′ (X t k ),P ′ (X t k+1 |X t k ) 1[X t k+1 ̸= X t k ] log Q ′ t k (X t k ,X t k+1 ) Q t k (X t k ,X t k+1 ) = K−1 X k=0 E P ′ (X t k ) X y̸=X t k P ′ (y|X t k ) log Q ′ t k (X t k ,y) Q t k (X t k ,y) = K−1 X k=0 E P ′ (X t k ) X y̸=X t k ∆tQ ′ t k (X t k ,y) log Q ′ t k (X t k ,y) Q t k (X t k ,y) +O(∆t 2 ) = ∆t→0 E X 0:T ∼P ′ Z T 0 X y̸=X t Q ′ t log Q ′ t Q t (X t ,y) (♦) Plugging these expressions for (⋆) and (♦) into (7.21), we get: KL(P ′ ∥P) = KL(π ′ 0 ∥π 0 ) | z (⋆) + E X 0:T ∼P ′ Z T 0 X y̸=X t Q ′ t log Q ′ t Q t (X t ,y)dt |z (♦) + E X 0:T ∼P ′ Z T 0 X y̸=X t (Q t −Q ′ t )(X t ,y)dt |z (⋄) = KL(π ′ 0 ∥π 0 ) + E X 0:T ∼P ′ X y̸=X t Q ′ t log Q ′ t Q t +Q t −Q ′ t (X t ,y) dt= KL(π ′ 0 ∥π 0 ) + E X 0:T ∼P ′ X y̸=X t Q ′ t log Q ′ t Q t +Q t −Q ′ t (X t ,y) dt (7.22) which is the simplified KL divergence betweenP ′ andP. We can also use the equalityQ t (x,x) = 1− P y̸=x Q t (x,y) to write P y̸=X t (Q t −Q ′ t )(X t ,y) = (Q ′ t −Q t )(X t ,X t ), which gives us: KL(P ′ ∥P) = KL(π ′ 0 ∥π 0 ) + E X 0:T ∼P ′ Z T 0 X y̸=X t Q ′ t log Q ′ t Q t (X t ,y) + Z T 0 (Q ′ t −Q t )(X t ,X t ) dt (7.23) which is an equivalent expression for KL divergence between CTMCs using the rate of remaining at a position or the stay rate.□ This decomposition reveals that the KL divergence between two CTMC path measures is the sum of the discrepancy between their initial marginals and a time-integrated KL divergence of the generator matrices at jump times, where the stateX t jumps to a new statey ̸=X t . This provides an intuitive interpretation for the discrete Schr ̈odinger bridge problem, which selects, among all Markov processes matching the marginals, the one whose jump dynamics deviates minimally in KL divergence from the reference dynamics. While we defined the canonical reference process in the continuous state space as Brownian motion with a reference driftf, which is often set tof:= 0, we can also define two common forms of the reference generator Q 0 below. Forms of the Reference Generator for CTMCs The reference generatorQ 0 defining the CTMCQis the baseline discrete stochastic dynamics that is minimally reweighted to match the prescribed marginal constraints. The 186 Figure 21: Discrete Schr ̈odinger Bridge Problem. The discrete SB problem seeks a controlled CTMC path measureP ⋆ that is closest, in KL divergence, to a reference CTMCQwhile matching the prescribed marginal distributionsπ 0 andπ T . Top: CTMC trajectories under the sub-optimal controlled path measureP u with generator Q u . Bottom: CTMC trajectories under the Schr ̈odinger bridge path measureP ⋆ with generatorQ ⋆ obtained by minimizing the KL divergence with the reference generator Q 0 while maintaining the terminal marginals. choice ofQ 0 fundamentally changes the solution to the (Discrete SB Problem). Here, we will highlight the uniform generator and the pre-trained generator as common choices. Uniform Generator. The uniform generator Q 0 t is defined as: Q 0 t (x,y) = ( γ(t) |X|−1 y ̸= x −γ(t) y = x (Uniform Generator) which corresponds to a homogeneous jump process such that from any statex∈Xat timet, the state remains unchanged with rate−γ(t) and jumps uniformly to any other statey ̸=x. This means that the optimal generatorQ ⋆ is entirely determined by the marginal constraints. It can be interpreted as the discrete state space analog of pure Brownian motion and is useful for settings where prior dynamics are unknown. Baseline Transition Generator. A more structured alternative is to define the reference generatorQ 0 t with baseline transition rates of the system’s natural dynamics or from a pretrained model, where the model is trained to capture the dynamics of the system from data samples. This choice embeds prior knowledge about plausible transitions into the reference process. The Schr ̈odinger bridge then acts as a minimal correction of the pretrained dynamics to match the prescribed marginals. While we have defined the (Discrete SB Problem) objective and provided common forms of the reference generator, it remains unknown how to optimize the generator such that it yields the optimal 187 CTMC while preserving the marginal constraints. In the remainder of Section 7, we introduce two concrete methods of solving the discrete SB problem, each of which has deep connections to methods in the continuous state space. Specifically, we introduce stochastic optimal control (SOC) of CTMCs, which extends the SOC ideas from Section 3 to the discrete state space, and discrete diffusion Schr ̈odinger bridge matching, which extends the ideas of Markov and reciprocal projections from Section 4.5 and diffusion Schr ̈odinger bridge matching algorithm from Section 6.3 to the discrete state space. 7.3 Stochastic Optimal Control of CTMCs Prerequisite: Section 3 Just like in the continuous state space, we can reframe the SB problem as a stochastic optimal control (SOC) problem, where the Schr ̈odinger bridge aligns with the lowest cost path from any intermediate state to a state in the target distribution. In this section, we build the theoretical foundations of SOC in the discrete state space, highlighting the deviations from the continuous state-spacee formulation, which will lead us to explicitly defining objectives for tractably solving the discrete SB problem with SOC in Section 7.4. First, we define the cost functionalJ(x,t;u) :X ×[0,T]→ Rwhich returns the cost-to-go from a intermediate state x∈X to the target distribution π T under the controlled path measure P u : J (x,t;u) := E X u 0:T ∼P u Z T 0 X y̸=X u s C t (X u s ,y)ds + Φ(X u T ) X u t = x (7.24) which measures the expected cumulative running costC t (x,y) incurred from timetuntil the terminal timeTand the terminal cost Φ(x) :X → R. The running cost is defined as the instantaneous KL divergence between the controlled generator Q u t and the reference generator Q 0 t given by: C t (x,y) := KL(p u t ∥q t ) = Q u t log Q u t Q 0 t −Q u t +Q 0 t (x,y)(7.25) The objective of the SOC problem is to determine the optimal control generatorQ ⋆ :=Q u ⋆ that minimizes the cost-to-go functional: J ⋆ (x,t;u ⋆ ) := inf P u J (x,t;u), ∀(x,t)∈X × [0,T ](7.26) which generates the optimal CTMC path measureP ⋆ :=P u ⋆ with generatorQ ⋆ t :=Q u ⋆ t . Analo- gous to the continuous setting, the optimal cost-to-go satisfies (Bellman’s Principle of Optimality). For a small time increment ∆t, the cost decomposes into the cost accumulated over [t,t+ ∆t] and the optimal cost from t + ∆t onward: J ⋆ (x,t;u ⋆ ) = inf P u E X u t:T ∼P u Z t+∆t t + Z T t+∆t X y̸=X u s C t (X u s ,y)ds + Φ(X u T ) X u t = x = inf P u ∆t X y̸=x C t (x,y) +O(∆t 2 ) |z optimal cost of jumps over t→t+∆t + E X u t+∆t:T ∼P u [J ⋆ (X u t+∆t ,t + ∆t)|X u t = x] |z optimal cost of all jumps over t+∆t→T (7.27) The optimally controlled measureP ⋆ can be obtained by defining the value functionV t as the optimal cost-to-go V t (x) := J ⋆ (x,t;u ⋆ ), which yields the dynamic programming relation: V t (x) = inf P u ∆t X y̸=x C t (x,y) +O(∆t 2 ) + E X u t+∆t:T ∼P u [V t+∆t (X u t+∆t )|X u t = x] (7.28) 188 Since no more running costC t can be incurred at timet=T, the terminal value function is equal to the terminal cost V T (x) = Φ(x). To obtain an explicit characterization of the optimal CTMC dynamics solving the stochastic optimal control problem in (7.27), we derive the structure of the optimal process in a sequence of steps. The key goal is to connect the dynamic programming formulation of the SOC problem with the change-of-measure perspective underlying Schr ̈odinger bridges. Concretely, we first determine the form of the optimal controlled generatorQ ⋆ , then characterize the value function through the Hamilton–Jacobi–Bellman equation, and finally use these results to recover the optimal path measure P ⋆ and its likelihood ratio with respect to the reference process. The derivation proceeds as follows: (i)We derive the form of the optimal generatorQ ⋆ t that defines the law of paths under the path measure P ⋆ that solves the SOC problem in (7.27) (Proposition 7.6). (i)We show that the value function in (7.28) satisfies the Hamilton-Jacobi-Bellman equation (Corollary 7.8). (iv)We derive the form of the optimal path measureP ⋆ that solves the SOC problem (Proposition 7.9). (i)Finally, we derive the Radon-Nikodym derivative (RND) between the optimal path measure P ⋆ , and the reference path measure Q (Proposition 7.10). Together, these theoretical ideas will form the basis for solving the discrete SB using SOC. We start with the derivation of the optimal generatorQ ⋆ t with respect to the value functionV t (7.28) which will naturally lead to the proof that the value function satisfies the HJB equation. Proposition 7.6 (Optimal Generator). Given the generatorQ 0 t of the reference processQ, the optimal generatorQ ⋆ t of the processP ⋆ that solves the SOC problem in (7.27) takes the form: Q ⋆ t (x,y) = Q 0 t (x,y)e V t (x)−V t (y) , ∀x,y ∈X(Optimal Generator) where V t :X → R is the value function defined in (7.28). Proof. First, we expand the second term of the value function in (7.28), which defines the Bellman recursion, to get: inf P u E X u t+∆t:T ∼P u V t+∆t (X u t+∆t ) X u t = x = inf P u " X y∈X V t+∆t (y)P u (X u t+∆t = y|X u t = x) # (7.4) = inf P u " X y∈X V t+∆t (y) 1 x=y + ∆tQ u t (x,y) +O(∆t 2 ) # = inf P u V t+∆t (x)(1 x=y + ∆tQ u t (x,x)) | z Case: y=x + X y̸=x V t+∆t (y)∆tQ u t (x,y) | z Case: y̸=x +O(∆t 2 )) (7.29) 189 For any CTMC generator, the sum of transition rates from a given statexto all statesy ∈Xmust sum to zero, we have Q t (x,x) =− P y̸=x Q t (x,y), which we can substitute into (7.29) to get: inf P u E X u t+∆t:T ∼P u V t+∆t (X u t+∆t ) X u t = x = inf P u V t+∆t (x) 1 x=y −∆t X y̸=x Q u t (x,y) + X y̸=x V t+∆t (y)∆tQ u t (x,y) +O(∆t 2 )) = inf P u V t+∆t (x)−∆t X x̸=y V t+∆t (x)Q u t (x,y) + ∆t X x̸=y V t+∆t (y)Q u t (x,y) +O(∆t 2 ) = V t+∆t (x) + ∆t inf P u X x̸=y Q u t (x,y)(V t+∆t (y)− V t+∆t (x)) +O(∆t 2 )(7.30) Substituting this back into (7.28), we have: V t (x) = inf P u ∆t X y̸=x C t (x,y) +O(∆t 2 )+V t+∆t (x) + ∆t X y̸=x Q u t (x,y)(V t+∆t (y)− V t+∆t (x)) +O(∆t 2 ) V t (x)− V t+∆t (x) = inf P u ∆t X y̸=x C t (x,y) + ∆t X y̸=x Q u t (x,y)(V t+∆t (y)− V t+∆t (x)) +O(∆t 2 ) V t+∆t (x)− V t (x) =− inf P u ∆t X y̸=x C t (x,y) + ∆t X y̸=x Q u t (x,y)(V t+∆t (y)− V t+∆t (x)) +O(∆t 2 )(7.31) where we rearrange the Bellman recursion to express the forward differenceV t+∆t (x)−V t (x), which introduces a minus sign before the infimum. This expression is used to get the time derivative∂ t V t , by dividing both sides by ∆t and take the limit ∆t→ 0: ∂ t V t (x) = lim ∆t→0 V t+∆t (x)− V t (x) ∆t =− inf P u X y̸=x C t (x,y) +Q u t (x,y)(V t (y)− V t (x)) |z f (Q u ) (7.32) Definingf(Q u t ) as the function inside the infimum for allx̸=yand expandingC t (x,y) as the KL divergence in (7.27), we can take the derivative f ′ (Q u t ) with respect to Q u t to get: f (Q u t ) = Q u t log Q u t Q 0 t (x,y)−Q u t (x,y) +Q 0 t (x,y) +Q u t (x,y)(V t (y)− V t (x))(7.33) f ′ (Q u t ) = log Q u t Q 0 t (x,y) + V t (y)− V t (x)(7.34) Setting f ′ (Q u t ) = 0 to obtain the minimizer Q ⋆ t : log Q ⋆ t Q 0 t (x,y) = V t (x)− V t (y) =⇒ Q ⋆ t (x,y) = Q 0 t (x,y)e V t (x)−V t (y) (7.35) which is the form of the optimal generator.□ From the result (7.35), we can also observe that by rewriting the exponential as a fraction, we recover a form analogous the Doob’s h-transform described in Section 4.4 for CTMCs. Remark 7.7 (Doob’sh-Transform of CTMCs). The optimal generatorQ ⋆ t is the Doob h-transform of the reference generatorQ 0 t where thehfunction is defined as the exponentiated 190 value function h(x,t) := e −V t (x) : Q ⋆ t (x,y) = Q 0 t (x,y) e V t (x) e V t (y) = Q 0 t (x,y) e −V t (y) e −V t (x) =: Q 0 t (x,y) h(y,t) h(x,t) (7.36) which can be interpreted as tilting the generator toward statesythat minimize the optimal cost-to-go defined by V t (x) := J ⋆ (x,t;u ⋆ ). Now we will show that the value function satisfies the HJB equation, which is analagous to our derivation in Section 2.7 in continuous state spaces. Crucially, this defines the discrete SOC problem with a non-linear PDE which can be transformed to a linear equation via exponentiation, which acts as a discrete analog of the Hopf-Cole transform discussed in Section 2.8. Corollary 7.8 (Hamilton-Jacobi-Bellman Equation). The value function in (7.28) satisfies the Hamilton-Jacobi-Bellman equations, defined as: ∂ t V t (x) = X y̸=x Q 0 t (x,y) e V t (x)−V t (y) − 1 ⇐⇒ ∂ t e −V t (x) = X y̸=x Q 0 t (x,y) e −V t (x) − e −V t (y) Proof. To prove this, we substitute the final form of the optimal generatorQ ⋆ t =Q 0 t e V t (x)−V t (y) defined in (Optimal Generator) into the equation for ∂ t V t in (7.32) to get: ∂ t V t (x) =− X y̸=x Q ⋆ t log Q ⋆ t Q 0 t −Q ⋆ t +Q 0 t +Q ⋆ t (V t (y)− V t (x)) =− X y̸=x Q 0 t e V t (x)−V t (y) log Q 0 t e V t (x)−V t (y) Q 0 t −Q 0 t e V t (x)−V t (y) +Q 0 t +Q 0 t e V t (x)−V t (y) (V t (y)− V t (x)) =− X y̸=x −Q 0 t e V t (x)−V t (y) (V t (y)− V t (x))−Q 0 t e V t (y)−V t (x) +Q 0 t +Q 0 t (x,y)e V t (x)−V t (y) (V t (y)− V t (x)) =− X y̸=x Q 0 t 1− e V t (x)−V t (y) = X y̸=x Q 0 t e V t (x)−V t (y) − 1 (7.37) which gives us the first HJB equation in the Corollary. To get the second expression, we can differentiate e −V t (x) and apply the chain rule to get: ∂ t e −V t (x) = e −V t (x) ∂ t V t (x) (7.37) = e −V t (x) X y̸=x Q 0 t (x,y) e V t (x)−V t (y) − 1 = X y̸=x Q 0 t (x,y) e −V t (y) − e −V t (x) (7.38) which is the second HJB equation in the Corollary.□ Using this Corollary, we can derive the optimal path measure, which can be defined in terms of the value function V t (x). Proposition 7.9 (Optimal Path Measure). The optimal path measureP ⋆ given the value function V t (x) can be expressed as: p ⋆ t (x) = 1 Z t q t (x)e −V t (x) , Z t := E x∼q t h e −V t (x) i (7.39) 191 Proof. From Remark 7.7, we can consider the probability of a statex ∈ Xunder the optimal path measure as its probability under the reference measure tilted by theh-function defined as h(x,t) :=e −V t (x) which yieldsξ t (x) = 1 Z q t (x)e −V t (x) , whereZis the normalization factor. To show thatξ(x) is indeed the optimal path measure, we must check that it satisfies Kolmogorov’s forward equation defined in Proposition 7.1 for the optimal generatorQ ⋆ t . First, taking the partial derivative, we get: ∂ t ξ t (x) = ∂ t 1 Z q t (x)e −V t (x) = 1 Z e −V t (x) ∂ t q t (x) + q t (x)∂ t e −V t (x) (7.40) Applying the Kolgomorov forward equation from Lemma 7.1 to the reference path measureQand the HJB equations from Corollary 7.8, we have: ∂ t p 0 t (x) (7.1) = X x̸=y Q 0 t (y,x)q t (y)−Q 0 t (x,y)q t (x) (7.41) ∂ t e −V t (x) (7.8) = X x̸=y Q 0 t (x,y) e −V t (x) − e −V t (y) (7.42) and substituting this back into (7.40), we get: ∂ t ξ t (x) = 1 Z e −V t (x) X x̸=y Q 0 t (y,x)q t (y)−Q 0 t (x,y)q t (x) + q t (x) X x̸=y Q 0 t (x,y) e −V t (x) − e −V t (y) = X x̸=y Q 0 t (y,x) 1 Z q t (y)e −V t (x) −Q 0 t (x,y) 1 Z q t (x)e −V t (x) +Q 0 t (x,y) 1 Z q t (x)e −V t (x) | z =0 −Q 0 t (x,y) 1 Z q t (x)e −V t (y) = X x̸=y Q 0 t (y,x) 1 Z q t (y)e −V t (y) | z =:ξ t (y) e V t (x)−V t (y) −Q 0 t (x,y) 1 Z q t (x)e −V t (x) | z =:ξ t (x) e V t (x)−V t (y) = X x̸=y Q 0 t (y,x)e V t (x)−V t (y) | z =:Q ⋆ t (y,x) ξ t (y)−Q 0 t (x,y)e V t (x)−V t (y) |z =:Q ⋆ t (x,y) ξ t (x) = X x̸=y (Q ⋆ t (y,x)ξ t (y)−Q ⋆ t (x,y)ξ t (x))(7.43) which is exactly the Kolmogorov forward equation for the generatorQ ⋆ t that defines the optimal path measureP ⋆ . Since we prove in Lemma 7.1 that the solution to the Kolmogorov forward equation is unique, we have shown thatp ⋆ t (x) = 1 Z q t (x)e −V t (x) . We deriveZ t such that the probability distribution is normalized, i.e. P x∈X p ⋆ t (x) = 1, as follows: X x∈X p ⋆ t (x) = 1 Z t X x∈X q t (x)e −V t (x) = 1 =⇒ Z t = X x∈X q t (x)e −V t (x) = E x∼q t h e −V t (x) i (7.44) which concludes our proof of the optimal path measure.□ Using this form of the optimal path measurep ⋆ t (x) = 1 Z q t (x)e −V t (x) , we can now derive the RND between the optimal and reference path measures. Proposition 7.10 (Radon-Nikodym Derivative of Optimal and Reference Path Measure). The Radon-Nikodym Derivative (RND) of the optimal path measureP ⋆ with generator Q ⋆ t and the reference path measure Q with generator Q 0 t is given by: dP ⋆ dQ (X 0:T ) = 1 Z e −Φ(X T ) , Z := E x∼q T h e −Φ(X T ) i (Optimal RND) 192 where Φ(x) :X → R is the terminal cost function. Proof. Starting from the definition of RND between two CTMC path measures in Proposition 7.4, we can write: log dP ⋆ dQ (X 0:T ) = log p ⋆ 0 (X 0 ) q 0 (X 0 ) + X t:X t − ̸=X t log Q ⋆ t (X t − ,X t ) Q 0 t (X t − ,X t ) + Z T 0 X y̸=X t (Q 0 t −Q ⋆ t )(X t ,y)dt(7.45) Now, we can substitute the expression for the optimal path probabilityp ⋆ 0 (x) = 1 Z q 0 (x)e −V 0 (x) from Proposition 7.9 and the optimal generator Q ⋆ (x,y) = Q 0 t (x,y)e V t (x)−V t (y) from Proposition 7.6 to get: log dP ⋆ dQ (X 0:T ) = log 1 Z 0 q 0 e −V 0 (X 0 ) q 0 (X 0 ) + X t:X t − ̸=X t log Q 0 t (X t − ,X t )e V t (X t − )−V t (X t ) Q 0 t (X t − ,X t ) + Z T 0 X y̸=X t Q 0 t (X t ,y)−Q 0 t (X t ,y)e V t (X t )−V t (y) dt =−V 0 (X 0 )− logZ 0 + X t:X t − ̸=X t (V t (X t − )− V t (X t )) + Z T 0 X y̸=X t Q 0 t (X t ,y) 1− e V t (X t )−V t (y) dt (7.46) Since a CTMC is a piecewise c`adl`ag function, we can define jump times 0 =t 0 < t 1 < · < t k < · < t K−1 < t K =T. Then, we have that in the time interval [t k ,t k+1 ] the stateX t k stays fixed until the left limitX t − k+1 where it jumps to stateX t k+1 . Therefore, we can define the value difference over all time steps as the sum of changes in value at stateX t k over the time interval [t k ,t k+1 ] and the value change over the jump between states X t k and X t k+1 , given by: V T (X T )− V 0 (X 0 ) = K−1 X k=0 V t k+1 (X t k )− V t k (X t k ) + K−1 X k=1 V t k (X t k )− V t k (X t k−1 ) = K−1 X k=0 Z t k+1 t k ∂ t V t (X t )dt + X t:X t − ̸=X t (V t (X t )− V t (X t − )) = Z T 0 ∂ t V t (X t )dt + X t:X t − ̸=X t (V t (X t )− V t (X t − ))(7.47) Isolating V 0 (X 0 ) and substituting the HJB equation from Corollary 7.8, we get the expression: −V 0 (X 0 ) =−V T (X T ) + Z T 0 ∂ t V t (X t )dt + X t:X t − ̸=X t (V t (X t )− V t (X t − )) =−V T (X T ) + Z T 0 X y̸=X t Q 0 t (X t ,y) e V t (X t )−V t (y) − 1 dt + X t:X t − ̸=X t (V t (X t )− V t (X t − )) (7.48) Finally, substituting (7.48) back into (7.46) and cancelling terms, we get: log dP ⋆ dQ (X 0:T ) =−V T (X T )− logZ 0 =⇒ dP ⋆ dQ (X 0:T ) = 1 Z 0 e −V T (X T ) (7.49) which yields the form of the RND between the CTMC that solves the SOC problem and the reference path measure. Substituting V T (X T ) = Φ(X T ) yields the final result.□ 193 Having derived the stochastic optimal control formulation for CTMCs, we have shown that the optimal dynamics arise from an exponential tilting of the reference generator and that the resulting optimal path measureP ⋆ admits a simple Radon–Nikodym derivative with respect to the reference CTMCQ. In particular, the likelihood ratio depends only on the terminal value functionV T (X T ), revealing that the SOC solution can be interpreted as an entropy-regularized change of measure on path space. From this perspective, solving the SOC problem is equivalent to computing a KL projection of path measures, where the optimal controlled process is the closest process to the reference dynamics that satisfies the desired terminal value constraint. This observation provides a direct connection to the Schr ̈odinger bridge problem. In the next section, we explicitly formulate discrete Schr ̈odinger bridges using the SOC framework, allowing the objectives developed in Section 3.3 to be adapted to discrete state spaces and enabling practical algorithms for learning optimal CTMC dynamics. 7.4 Discrete Schr ̈odinger Bridges with Stochastic Optimal Control Prerequisite: Section 3.3 Given our construction of SOC for CTMCs in Section 7.3, we can now explicitly write the (Discrete SB Problem) in the form of a stochastic optimal control functional, which contains a running cost that corresponds to the KL divergence between the controlled bridge measureP u and the reference measureQand a terminal cost that ensures the optimal process satisfies the terminal constraint X T ∼ π T . Definition 7.11 (Discrete Schr ̈odinger Bridge Problem with SOC). Consider the discrete SB problem whereQ 0 t is the generator of the reference CTMCQandπ 0 ,π T ∈P(X) are the initial and terminal constraints on the finite state spaceX. The discrete Schr ̈odinger bridge problem can be formulated as a stochastic optimal control (SOC) problem which seeks a controlled generator Q u t of a controlled CTMC P u that minimizes: inf P u E X u 0:T ∼P u Z T 0 X y̸=X u t C t (X u t ,y)dt + Φ(X u T ) , X u 0 ∼ π 0 (Discrete SB-SOC Objective) where the running costC t (x,y) :X ×X → Rand terminal cost Φ(x) :X → Rare defined as: C t (X u t ,y) := KL(p u t ∥q t ) = Q u t log Q u t Q 0 t −Q u t +Q 0 t (X u t ,y),Φ(X u T ) = log ˆφ(X u T ) π T (X u T ) (7.50) The optimal controlled processP ⋆ defines the Schr ̈odinger bridge betweenπ 0 andπ T relative to the reference dynamics Q 0 t . Now that we have established that the SOC problem can be solved in the discrete state space, we can easily adapt the objectives defined in Section 3.3 to solve the SB-SOC problem for discrete variables (N ̈usken et al., 2021; Zhu et al., 2025). Definition 7.12 (Discrete Relative Entropy (RE) Loss). The relative entropy (RE) loss between the controlled path measureP u and the optimal path measureP ⋆ is defined as the KL divergence: L RE (P u , P ⋆ ) := KL(P u ∥P ⋆ ) = E P u log dP u dP ⋆ = E P u log dP u dQ − log dP ⋆ dQ (Discrete RE Objective) 194 which can be written in terms of the controlled and reference generator Q u and Q 0 as: L RE (Q u ) := E X u 0:T ∼P u Z T 0 X y̸=X u t Q u t log Q u t Q 0 t +Q 0 t −Q u t (X u t ,y) | z log dP u dQ +V T (X u T ) + logZ 0 | z − log dP ⋆ dQ (Discrete RE Loss) As discussed in Section 3.3, the (Discrete RE Objective) requires backpropagating through the full stochastic trajectory simulations, so we can use (RERF Loss) as a practical surrogate loss for (Discrete RE Objective). Next, we will adapt the (CE Loss) to the discrete state space. Similarly to Definition 3.15, the expectation over the optimal path measure P ⋆ is generally intractable during training, so we define the expectation over an arbitrary path measureP v and reweighting the log RND by the RND between P ⋆ and P v . Definition 7.13 (Cross-Entropy Objective). The cross-entropy (CE) loss that optimizes a controlled path measureP u to match the optimal path measureP ⋆ is defined as the reverse KL divergence: L CE (P u , P ⋆ ) := KL(P ⋆ ∥P u ) = E P ⋆ log dP ⋆ dP u = E P v dP ⋆ dP v log dP ⋆ dP u = E P v dP ⋆ dP v log dP ⋆ dQ |z constant w.r.t. P u + log dQ dP u = E P v dP ⋆ dP v log dQ dP u + C(7.51) wherelog dP ⋆ dQ vanishes in the gradient as it is an additive constant with respect toP u . The CE objective can be written in terms of the controlled and reference generatorQ u andQ 0 as: L CE (Q u ) = 1 Z E X v 0:T ∼P v e W (X v 0:T ) Z T 0 X y̸=X v t Q 0 t log Q 0 t Q u t +Q u t −Q 0 t (X t ,y) + C(7.52) where W (X v 0:T ) := log dP ⋆ dP v (X v 0:T ) is a weight that can be expanded as: W (X v 0:T ) := log dP ⋆ dP v (X v 0:T ) = log dP ⋆ dQ (X v 0:T )− dP v dQ (X v 0:T ) = Z T 0 X y̸=X v t Q v t log Q v t Q 0 t +Q 0 t −Q v t (X v t ,y) + V T (X v T ) + logZ 0 (7.53) The sampling law is commonly defined in practice as the stop-gradient sampling measureP ̄u like in L RERF , where Q ̄u := stopgrad(Q u ) is the non-gradient-tracking controlled generator. The final objective introduced for continuous SOC in Section 3.3 is the variance and log-variance objectives. Recall that the RND between the optimal and controlled path measure is high when the two distributions are dissimilar, in the sense that paths with high probability underP ⋆ have a low probability underP u and vice versa. On the other hand, the variance is low when the distributions are similar and is minimized at zero exactly when the RND evaluates to a constant regardless of the stochastic path, indicating that the two measures are equal. Since the variance objective is generally unstable, we only define the log-variance objective below. 195 Definition 7.14 (Log-Variance Objectives). The log-variance (LV) loss that optimizes a controlled path measure P u to match the optimal path measure P ⋆ is defined as: L LV (P u , P ⋆ ) := Var P v log dP ⋆ dP u = Var P v log dP ⋆ dQ − log dP u dQ (7.54) which can be written in terms of the controlled and reference generator Q u and Q 0 as: L LV (Q u ) = Var X v 0:T ∼P v Z T 0 X y̸=X v t Q u t log Q u t Q 0 t +Q 0 t −Q u t (X v t ,y) | z log dP u dQ (X v 0:T ) +V T (X v T ) + logZ 0 | z − log dP ⋆ dQ (X v 0:T ) (7.55) These objectives allow us to tractably solve the (Discrete SB-SOC Objective) by simulating the controlled CTMCP u over discrete time steps while simultaneously evaluating the transition rates of each jump under the reference generatorQ 0 , computing the SOC objective using the time-discretized form of the log RND, and optimizinguusing standard gradient descent. For uniform discrete time steps 0 =t 0 < · < t k < · < t K =Twith increments ∆t, the log RND over the discrete states (X t k ) k∈0,...,K used to compute the losses can be computed as: log dP u dQ ((X t k ) k∈0,...,K ) = K−1 X k=0 ∆t Q u t k log Q u t k Q 0 t k +Q 0 t k −Q u t k (X t k ,X t k+1 ) +O(∆t 2 ) (Discretized Log RND) Then, we can define a general form of the (Discrete SB-SOC Objective) training procedure (Tang et al., 2025b; Zhu et al., 2025) below: Discrete SB-SOC Training Framework Training a parameterized generatorQ u using one of the SOC objectives defined above typically iterates through the following steps: (i) Sample descretized trajectories (X v t k ) t∈0,...,K using the proposal generatorQ v forK steps. Alternatively, if the proposal generator is notuand trajectories can be reused, then samples can be stored and subsequently sampled from a replay buffer B. (i)For each trajectory, compute the transition ratesQ 0 t k (X t k ,X t k+1 ) andQ u t k (X t k ,X t k+1 ) over every interval [t k ,t k+1 ]. (i)Compute the (Discretized Log RND) for each trajectory and use it to compute one of the SOC losses L(Q u ). (i) Optimize the parameterized generator Q u using gradient steps of ∇ u L(Q u ) (iv) Repeat from Step (i). In practice, the proposal generatorQ v is commonly defined as the same generator being optimized Q u but without gradient-tracking, i.e., Q v := Q ̄u = stopgrad(Q u ). The SOC framework provides a practical procedure for learning the controlled generatorQ u of a CTMC through sampling discrete trajectories from a proposal generatorQ v and optimizing the SOC objective with respect toQ u . While this process is well-suited for tasks where we have no access to samples from the target distributionπ T but only a way of evaluating the likelihood of a sample under it, optimizing a Schr ̈odinger bridge on paired samples or an explicit optimal transport map requires a different approach. Next, we will describe the extension of Markovian and reciprocal projections in 196 the discrete state space, which allows us to optimize discrete Schr ̈odinger bridges directly on samples from the initial and target distributions, or pairs from an optimal transport coupling π ⋆ 0,T . 7.5 Discrete Markov and Reciprocal Projections Prerequisite: Section 4.5 In this section, we extend the theory of Markov and reciprocal projections to discrete state spaces, where stochastic processes are characterized by the generators of continuous-time Markov chains (CTMCs). Recall from Section 4.5 that the Markovian projection identifies the Markov process that is closest in KL divergence to a given reciprocal process within the class of Markov dynamics, while the reciprocal projection enforces the endpoint constraints through conditioning on the boundary distributions. M ⋆ := proj M (Π) = arg min M∈M KL (Π∥M)(Markovian Projection) Π ⋆ := Q ·|0,T M 0,T = proj R(Q) (P) = Z R d ×R d Q ·|0,T (·|x 0 ,x T )dM 0,T (x 0 ,x T )(Reciprocal Projection) Since this section is quite notation-heavy, we will first establish the meaning behind some and shorthand for the notation. Notation for Section 7.5 and 7.6 NotationMeaning Qreference CTMC path measure Q 0 t (x,y)generator of the reference process Q MMarkov CTMC path measure in the Markov class M ΠCTMC path measure in the reciprocal class R(Q) M t , Π t marginal distributions at timetunder a Markov or reciprocal path measure M ⋆ Markov projection of a reciprocal bridge measure Π Q M ⋆ (x,y)generator of Markovian projection M ⋆ Π ⋆ reciprocal projection of a Markov measure M Π ·|0 = Π ·|0=x 0 mixture of bridges conditioned on initial stateX 0 =x 0 . Transitions under the conditioned measure, denotedQ ·|0=x 0 t|s (y|x), are also conditioned on x 0 Q Π ·|0 (x,y) = Q Π ·|0=x 0 (x,y) generator of reciprocal process conditioned on some initial state X 0 = x 0 Q ·|T = Q ·|T =x T reference path measure conditioned on terminal stateX T =x T . Transitions under the conditioned measure, denotedQ ·|T =x T t|s (y|x) are also conditioned on x T Q 0 t (x,y;x T )generator of the reference processQconditioned on the terminal state X T = x T Q ·|0 = Q ·|0=x 0 reference path measure conditioned on terminal stateX 0 =x 0 . Transitions under the conditioned measure, denotedQ ·|0=x 0 t|s (y|x) are also conditioned on x 0 ̃ Q 0 s (x,y;x 0 ) reverse-time generator of the reference processQconditioned on the state X 0 = x 0 197 Q t|s (y|x), Π t|s (y|x)transition density from statexat timesfrom stateyat timet under the specified path measure. Shorthand forQ t|s (X t =y|X s = x), Π t|s (X t = y|X s = x). We will derive the explicit form of the Markovian and reciprocal projections in terms of the generators of CTMCs, which will serve as the theoretical basis for extending the iterative Markovian fitting (IMF) procedure from Section 6.3 to the finite state space. (i)First, we define the Markovian projectionM ⋆ of a CTMC given a bridge measure in the reciprocal class Π∈ R(Q). We define the explicit form of its generatorQ M ⋆ and the KL divergence with the reciprocal process (Definition 7.15). (i)We derive the definition for the reference generatorQ 0 t conditioned on a terminal statex T , which appears in the expression for the generator of the Markovian projection (Lemma 7.16). (i) We show that conditioning the bridge measure in the reciprocal class Π∈R(Q) on an initial state X 0 = x 0 yields a Markov measure, which we denote Π ·|0=x 0 (Lemma 7.17). (iv)We derive the form of the KL divergenceKL(Π∥M) using the generator of the conditioned reciprocal process Q Π ·|0=x 0 t and the generator of the Markov process M (Lemma 7.18). (v) Finally, we define the reverse-time Markovian projection, which will allow us to condition on both the initial distributionπ 0 and target distributionπ T , reducing error accumulation during the IMF procedure (Definition 7.19). We start by defining the Markovian projectionM ⋆ of a CTMC path measure, its explicit form in terms of the generator matrices and the KL minimization objective that yieldsM ⋆ given a reciprocal measure Π. Definition 7.15 (Markovian Projection of CTMC Path Measure). Consider a reference path measureQwith generatorQ 0 t and a measure Π∈ R(Q) in the reciprocal class ofQthat preserves the bridge. The Markovian projectionM ⋆ :=proj M (Π) has a generatorQ M ⋆ t defined as the expectation over conditional generatorsQ 0 t (·,·;x) under the reference process Q of the form: Q M ⋆ t (x,y) = E x T ∼Q T|t (·|x) Q 0 t (x,y;x T ) X t = x (Generator of Markovian Projection) where each conditional generator is defined as: Q 0 t (x,y;x T ) = Q 0 t (x,y) Q T|t (x T |y) Q T|t (x T |x) − 1 x=y X z∈X Q 0 t (x,z) Q T|t (x T |z) Q T|t (x T |x) (Endpoint-Conditioned Generator) whereQ T|t (·|·) is the conditional transition probability under the reference measureQ. We can also define the KL divergence between Π and its Markovian projectionM ⋆ using Corollary 7.5 as: KL(Π∥M ⋆ ) = Z T 0 E Π 0,t X y̸=X t Q Π ·|0 t log Q Π ·|0 t Q M ⋆ t +Q M ⋆ t −Q Π ·|0 t ! (X t ,y) dt(7.56) where the KL divergence between the initial distributions vanishes as we assume they are aligned. Specifically, we define Π ·|0=x 0 as the conditional bridge measure with generator 198 Q Π ·|0=x 0 t defined as: Q Π ·|0=x 0 t (x,y) = E x T ∼Π T|0,t Q 0 t (x,y;x T )|X 0 = x 0 ,X t = x (7.57) where for any t∈ [0,T ], the marginal distributions match M t = Π t . Breaking down Definition 7.15, we introduce several unfamiliar definitions, including the endpoint- conditioned generator of the reference processQ 0 t (x,y;x T ), the reciprocal bridge measure Π conditioned onX 0 =x 0 denoted Π ·|0=x 0 and its generator defined asQ Π ·|0=x 0 t . In the following sequence of Lemmas, we will break down these ideas more concretely to better understand the discrete analogue of the Markovian projection. Lemma 7.16 (Conditional Generator of Markov Process). Consider a CTMCX 0:T under the reference path measureQwith generatorQ 0 t . ConditioningQon a terminal stateX T =x T gives the conditioned path measure, denotedQ ·|T =x T , which is Markov and is defined by the generator: Q 0 t (x,y;x T ) = Q 0 t (x,y) Q T|t (x T |y) Q T|t (x T |x) − 1 x=y X z∈X Q 0 t (x,z) Q T|t (x T |z) Q T|t (x T |x) (7.58) Proof. To confirm the Markov property of Q ·|T =x T , we first apply Bayes’ rule 31 to get: Q t|s,T (y|x,x T ) = Q t|s (y|x)Q T|t (x T |y) Q T|s (x T |x) |z Bayes’ rule =: Q ·|T =x T t|s (y|x)(7.59) which defines the transition kernel of the bridge process conditioned onX T =x T . This shows that the conditioned process is still Markov, that is given the current state X T = x T and the fixed endpointX T =x T , the law of the future stateX t depends on the past only through the current state X s . Now, applying the Kolmogorov forward equation from Lemma 7.1 and the Kolmogorov backward equation from Lemma 7.2, we obtain the generator of the conditioned process as: Q 0 t (x,y;x T ) = ∂ s Q ·|T =x T t|s (y|x) s=t = ∂ s Q t|s (y|x)Q T|t (x T |y) Q T|s (x T |x) s=t = ∂ s Q t|s (y|x) Q T|t (x T |y) Q T|t (x T |x) + Q t|t (y|x) |z 1 x=y ∂ t Q T|t (x T |y) Q T|t (x T |x) = Q 0 t (x,y) Q T|t (x T |y) Q T|t (x T |x) + 1 x=y − P z∈X Q t (x,y)Q T|t (x T |z) Q T|t (x T |x) = Q 0 t (x,y) Q T|t (x T |y) Q T|t (x T |x) − 1 x=y X z∈X Q 0 t (x,z) Q T|t (x T |z) Q T|t (x T |x) (7.60) which recovers for the form of the generatorQ 0 t (x,y;x T ) given the terminal stateX T =x T under the reference process Q.□ Next, we will derive the generator for the reciprocal process Π∈ R(Q) conditioned on the initial stateX 0 =x 0 , which is Markov as shown in Proposition 4.11. This allows us to define the KL 31 The standard form of Bayes’ rule statesp(y|z) = p(z|y)·p(y) p(z) , which we extend to add conditioning on a third variable x. 199 divergence between Π and its Markovian projectionM ⋆ =proj M (Π) which is necessary for optimizing the discrete SB. Lemma 7.17 (KL Divergence Between Reciprocal and Markov CTMCs). Given the reciprocal process Π∈ R(Q), conditioning on an initial stateX 0 =x 0 , yields a Markov process Π ·|0=x 0 that is defined by the generator: Q Π ·|0=x 0 t (x,y) = E x T ∼Π T|0,t Q 0 t (x,y;x T )|X 0 = x 0 ,X t = x (7.61) Then, for any Markov measureM∈MwhereM 0 = Π 0 and Π≪ M, the KL divergence between Π and M is defined as: KL(Π∥M) = E Π 0 h KL(Π ·|0=x 0 ∥M ·|0=x 0 ) i (7.62) whereKL(Π ·|0=x 0 ∥M ·|0=x 0 ) is a KL divergence between the Markov conditioned reciprocal process Π ·|0=x 0 and the conditioned Markov measure M ·|0=x 0 , which expands to: KL(Π ·|0 ∥M ·|0 ) = Z T 0 E Π 0,t X y̸=X t Q Π ·|0 t log Q Π ·|0 t Q M ·|0 t ! (X t ,y) + Q Π ·|0 t −Q M ·|0 t (X t ,X t ) dt (7.63) Proof. We prove each part of the Lemma in steps. First, we derive the generator of the conditioned reciprocal measure, which takes the form of an expectation of the conditional reference generators derived in Lemma 7.16. Step 1: Derive the Conditioned Generator. First, we establish that the pinned down bridge measure Π ·|0=x 0 is in the reciprocal class R(Q) and satisfies the (Reciprocal Property): Π ·|0=x 0 = Z Q ·|0,T (·|x 0 ,x T )Π T|0 (dx T |x 0 )(Pinned-Down Bridge) We first write the transition probability between states under the conditional process using the law of total probability: Π ·|0=x 0 t|s,0 (y|x) = X x T ∈X Π ·|0=x 0 T|s (x T |x)Π ·|0=x 0 t|s,T (y|x,x T )(Law of Total Probability) which samples the terminal stateX T =x T and then samples the intermediate timeX t =yfrom the bridge. Given that Π∈R(Q) is in the reciprocal class with a shared bridge measure asQ, by the (Reciprocal Property), the second term in the sum is equal to: Π ·|0=x 0 t|s,T (y|x,x T ) = Q t|s,T (y|x,x T ) = Q t|s (y|x)Q T|t (x T |y) Q T|s (x T |x) | z Bayes’ rule (7.64) Then, substituting this back into the expression for Π ·|0=x 0 t|s,0 (y|x) and recognizing Π ·|0=x 0 T|s (x T |x) = Π T|s,0 (x T |x,x 0 ), we have: Π ·|0=x 0 t|s,0 (y|x) = X x T ∈X Π ·|0=x 0 T|s (x T |x) Q t|s (y|x)Q T|t (x T |y) Q T|s (x T |x) = Q t|s (y|x) X x T ∈X Π T|s,0 (x T |x,x 0 ) |z (⋆) Q T|t (x T |y) Q T|s (x T |x) (7.65) 200 We can break down (⋆) using Bayes’ rule and applying the reciprocal property to get: Π T|s,0 (x T |x,x 0 ) = Π s|0,T (x|x 0 ,x T )Π T|0 (x T |x 0 ) Π s|0 (x|x 0 ) (Bayes’ Rule) = Q s|0,T (x|x 0 ,x T )Π T|0 (x T |x 0 ) Π s|0 (x|x 0 ) (Reciprocal Property) = Q s|0 (x|x 0 )Q T|s (x T |x) Q T|0 (x T |x 0 ) Π T|0 (x T |x 0 ) Π s|0 (x|x 0 ) (Bayes’ Rule) = Q s|0 (x|x 0 ) Q T|0 (x T |x 0 ) Π T|0 (x T |x 0 ) Π s|0 (x|x 0 ) Q T|s (x T |x)(7.66) Substituting (⋆) this back into (7.65), we can cancel terms and factor out terms not dependent on x T to get: Π ·|0=x 0 t|s (y|x) = Q t|s (y|x) X x T ∈X Q s|0 (x|x 0 ) Q T|0 (x T |x 0 ) Π T|0 (x T |x 0 ) Π s|0 (x|x 0 ) Q T|s (x T |x) Q T|t (x T |y) Q T|s (x T |x) = Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) Q t|s (y|x) X x T ∈X Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) Q T|t (x T |y)(7.67) To derive the generator using (7.67), we take the time derivative evaluated at t = s to get: Q Π ·|0=x 0 s (x,y) = ∂ t Π ·|0=x 0 (X t = y|X s = x) t=s (7.67) = ∂ t Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) Q t|s (y|x) X x T ∈X Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) Q T|t (x T |y) t=s = Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) ∂ t Q t|s (y|x) t=s X x T ∈X Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) Q T|t (x T |y) + Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) Q s|s (y|x) X x T ∈X Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) ∂ t Q T|t (x T |y) t=s (7.68) Applying the backward Kolmogorov equation from Lemma 7.2 to ∂ t Q T|t (x T |y), we have: Q Π ·|0=x 0 s (x,y) = Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) Q 0 t (x,y) X x T ∈X Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) Q T|t (x T |y) + Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) 1 x=y X x T ∈X " Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) − X z∈X Q 0 t (y,z)Q T|s (x T |z) !# = Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) Q 0 t (x,y) X x T ∈X Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) Q T|t (x T |y) −1 x=y Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) X x T ∈X " X z∈X Q 0 t (y,z)Q T|s (x T |z) Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) # (7.69) 201 Factoring out Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) and Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) and recognizing the (Endpoint-Conditioned Generator) of the Markov process defined in Lemma 7.16, we get: Q Π ·|0=x 0 s (x,y) = X x T ∈X Q s|0 (x|x 0 ) Π s|0 (x|x 0 ) Q 0 t (x,y)Q T|t (x T |y)− 1 x=y X z∈X Q 0 t (y,z)Q T|s (x T |z) | z =:Q 0 s (x,y;x T )Q T|s (x T |x) Π T|0 (x T |x 0 ) Q T|0 (x T |x 0 ) = X x T ∈X Q s|0 (x|x 0 ) Q T|0 (x T |x 0 ) Π T|0 (x T |x 0 ) Π s|0 (x|x 0 ) Q T|s (x T |x) |z =:Π ·|0=x 0 (X T =x T |X s =x) Q 0 s (x,y;x T ) = X x T ∈X Π ·|0=x 0 (X T = x T |X s = x)Q 0 s (x,y;x T ) = E x T ∼Π T|0,t Q 0 t (x,y;x T )|X 0 = x 0 ,X t = x (7.70) which gives us our final form for the conditional generator.□ Step 2: Derive the KL Divergence. In Corollary 7.5, we derived the expression for the KL divergence between CTMCs, which relies on the Markov structure. To derive the KL divergence between a non-Markov repciprocal process Π and a Markov measureM, we apply the (KL Divergence Chain Rule) from Lemma 1.4 to decompose the KL divergence as the sum of the KL divergence of the initial marginal and the expectation of the KL divergence of the conditioned measures: KL(Π∥M) = KL(Π 0 ∥M 0 ) |z =0 (Π 0 =M 0 ) + E Π 0 h KL(Π ·|0=x 0 ∥M ·|0=x 0 ) i = E Π 0 h KL(Π ·|0=x 0 ∥M ·|0=x 0 ) i (7.71) where the KL between the initial marginals vanishes by our definition Π 0 =M 0 . We have shown in Proposition 4.11 that conditioning a reciprocal process on an endpointX 0 =x 0 yields a Markov process and the KL divergence between two Markov measures is defined in Corollary 7.5, we can further expand (7.71) as: KL(Π∥M) (7.20) = Z T 0 E Π 0,t X y̸=X t Q Π ·|0 t log Q Π ·|0 t Q M ·|0 t ! (X t ,y) + Q Π ·|0 t −Q M ·|0 t (X t ,X t ) ds(7.72) which is exactly the form of the KL divergence between the reciprocal and Markov measures defined in the Proposition. Lemma 7.18 (Generator of Markovian Projection). The Markovian projectionM ⋆ =proj M (Π) of the reciprocal measure Π∈R(Q) that minimizes the KL divergenceKL(Π∥M) is defined by the generator: Q M ⋆ t (x,y) = E x T ∼Π T|t Q 0 t (x,y;x T )|X t = x (Markovian Projection Generator) Intuition. The goal is to find the closest Markov process to the reciprocal measure Π∈R(Q), which is not Markov and depends on the future stateX T . Since Markov measures can only depend on the present, we can remove the dependence onX T by taking the expectation over the conditional distribution of the endpoint. This yields the following candidate form of the Markov generator: Q M t (x,y) = E x T ∼Π T|t Q 0 t (x,y;x T )|X t = x (7.73) Now, we will show that this is indeed the generator of the Markovian projectionM ⋆ which minimizes the KL divergence with Π. 202 Proof. First, we assume that for allx̸=y,x T ∈Xandt∈[0,T), we haveQ 0 (x,y;x T ) = 0⇐⇒ Q 0 t (x,y). To prove that (7.73) is the generator of M ⋆ , we first decompose it as follows: Q M t (x,y) = E x T ∼Π T|t Q 0 t (x,y;x T )|X t = x = X x T ∈X Π T|t (x T |x)Q 0 t (x,y;x T )(7.74) Since the reciprocal process depends on both the initial state in addition to the target state, we apply the law of total expectation to insert conditioning on X 0 = x 0 to get: Q M t (x,y) = X x T ∈X X x 0 ∈X Π 0,T|t (x 0 ,x T |x)Q 0 t (x,y;x T ) = X x 0 ∈X Π 0|t (x 0 |x) X x T ∈X Π T|0,t (x T |x 0 ,x)Q 0 t (x,y;x T ) |z =:E Π T|0,t [Q 0 t (x,y;x T )|X 0 =x 0 ,x] = X x 0 ∈X Π 0|t (x 0 |x)Q Π ·|0=x 0 t (x,y) = E Π 0|t h Q Π ·|0=x 0 t (X t ,y) X t = x i (7.75) and we have shown that the generatorQ M t (x,y) is the expectation of bridge generators conditioned on the initial state or a mixture of bridges evaluated at x. Step 2: Show the Markov Generator Minimizes the KL Divergence. LetM∈Mbe an arbitrary Markov measure with generator defined in (Markovian Projection Generator), then by optimality ofM ⋆ , we haveKL(Π∥M)≥ KL(Π∥M ⋆ ). We aim to show that this inequality reduces to an equality where KL(Π∥M) = KL(Π∥M ⋆ ). Using the definition ofKL(Π∥M) from (7.63) and some algebra, we can write the differenceKL(Π∥M ⋆ )− KL(Π∥M) as: KL(Π∥M ⋆ )− KL(Π∥M) = E Π 0 Z T 0 X y̸=X t Q Π ·|0=x 0 log Q M t Q M ⋆ t +Q M ⋆ t −Q M t (X t ,y)dt = Z T 0 X y̸=X t E Π 0|t h Q Π ·|0=x 0 X t i | z (⋆) log Q M t Q M ⋆ t +Q M ⋆ t −Q M t (X t ,y)dt = Z T 0 X y̸=X t Q M t log Q M t Q M ⋆ t +Q M ⋆ t −Q M t (X t ,y)dt = KL(M∥M ⋆ )≥ 0(7.76) SinceM ⋆ is optimal, by definitionKL(Π∥M ⋆ )− KL(Π∥M)≤0. However, we have shown that KL(Π∥M ⋆ )− KL(Π∥M) =KL(M∥M ⋆ ) whereKL(M∥M ⋆ )≥0 by non-negativity of the KL divergence. Therefore, we can conclude that KL(Π∥M ⋆ ) = KL(Π∥M) and M = M ⋆ almost surely.□ Now, we have seen that the Markovian projectionM ⋆ :=proj R(Q) (Π) which minimizes the KL divergence from a reciprocal measure Π is explicitly defined with (Generator of Markovian Projection). However, recall from our discussion of Iterative Markovian Fitting from Section 6.3 that simulating the Markov SDE in the forward-time can accumulate errors in matching the target distributionπ T , motivating the simulation a corresponding reverse-time Markovian projection. Therefore, we will also extend the idea of a reverse-time Markovian projection for CTMCs. Definition 7.19 (Reverse-Time Markovian Projection). The generator of the reverse-time Markovian projection ̃ M ⋆ is given by: ̃ Q M ⋆ t (x,y) = E x 0 ∼Q 0|t (·|x) h ̃ Q 0 t (x,y;x 0 ) ̃ X t = x i (Reverse-Time Markov Generator) 203 where each conditional generator is defined as: ̃ Q 0 t (x,y;x 0 ) = ∂ s Q ·|0=x 0 (X t = y|X s = x) s=t = Q 0 T−t (x,y) Q T−t|0 (y|x 0 ) Q T−t|0 (x|x 0 ) − 1 x=y X z∈X Q 0 T−t (z,x) Q T−t|0 (z|x 0 ) Q T−t|0 (y|x 0 ) (7.77) whereQ T−t|0 (·|·) =Q(X T−t =·|X 0 =·) is the conditional transition probability under the reference measureQ. Then, the reverse-time KL divergence that is minimized under the Markovian projection M ⋆ is given by: KL(Π∥M ⋆ ) = Z T 0 E Π t,T X y̸=X t ̃ Q Π ·|T t log ̃ Q Π ·|T t ̃ Q M ⋆ t ! ( ̃ X t ,y) + ̃ Q M ⋆ t − ̃ Q Π ·|T t ( ̃ X t , ̃ X t ) dt (7.78) where the KL divergence between the terminal distributions at timeTvanishes as we initialize M T = Π T . Specifically, we define Π ·|T is the bridge measure conditioned onX T =x T with the reverse-time generator ̃ Q Π ·|T =x T t defined as: ̃ Q Π ·|T =x T t (x,y) = E x T ∼Π T|0,t h ̃ Q 0 t (x,y;x T )| ̃ X t = x, ̃ X 0 = x T i (7.79) where for any t∈ [0,T ], the marginal distributions match M ⋆ t = Π t . Proof Sketch. The result follows from the same proof sequence as the forward-time Markovian projection, which consists of deriving the form of the conditional generator in reverse-time coordinate (Lemma 7.16), showing that the reciprocal process conditioned on an initial state ̃ X 0 =x T is Markov (Lemma 7.17), and deriving the form of the optimal reverse-time generator of the Markovian projection (Lemma 7.18). This analogous proof works because the probability of a path is the same under time reversal, and the only difference is the direction of the generator. Therefore, by simply defining the reverse-time generator on the time-reversed CTMC process ̃ X t :=X T−t as in (Reverse-Time Markov Generator), we can compute the KL divergence between the generators of the reciprocal process Π ·|T conditioned on a state from timeTand the Markov processMevaluated along the reversed trajectory ̃ X 0:T . Taking expectation with respect to the trajectories from the reversed path measure ̃ X 0:T ∼Π yields the reverse KL divergence.□ Now that we have written the Markovian and reciprocal projections explicitly with respect to the generators of CTMCs, we can extend the Iterative Markovian Fitting (IMF) procedure described in Section 6.3 to the discrete state space, where instead of parameterizing the forward and reverse Markov drifts, we parameterize the forward and reverse Markov generators Q M θ and Q M φ . 7.6 Discrete Diffusion Schr ̈odinger Bridge Matching Prerequisite: Section 4.5, 6.3 Using these theoretical foundations of Markov and reciprocal projections of CTMCs, we can define the discrete state space analog of the Iterative Markovian Fitting (IMF) algorithm from Section 6.3 called the Discrete Diffusion Schr ̈odinger Bridge Matching (DDSBM) (Kim et al., 2024). Just like the IMF algorithm, DDSBM alternates between performing a Markovian projection to obtain a Markov generator and reciprocal projections to preserve the marginal constraints. Concretely, the algorithm is outlined as follows. 204 Discrete Diffusion Schr ̈odinger Bridge Matching DDSBM generates a sequence of Markov and reciprocal measures (M n ,Π n ) n∈N initialized at Π 0 := π 0,T Q ·|0,T by alternating between the following steps: (1a) Solve the forward-time Markovian projectionM 2n+1 :=proj M (Π 2n ) by updating a parameterized generator Q M θ to minimize L DDSBM (θ) := KL(Π 2n ∥M θ ) (1b) Define the reciprocal projection as Π 2n+1 := M 2n+1 Q ·|0,T (2a) Solve the backward-time Markovian projectionM 2n+2 :=proj M (Π 2n+1 ) by updating a parameterized generator ̃ Q M φ to minimize L DDSBM (φ) := KL(Π 2n+1 ∥M φ ) (2b) Define the reciprocal projection as Π 2n+2 := M 2n+2 Q ·|0,T While the high-level form of this algorithm is straightforward, we will build a deeper understanding of how the algorithm is implemented in practice by analyzing each step below. Step 1a: Forward-Time Markovian Projection. To obtain the Markovian projectionM 2n+1 = proj M (Π 2n ), we minimize the KL divergenceKL(Π 2n ∥M θ ) derived in Lemma 7.17 between the reciprocal measure from the previous iteration Π 2n = M φ 0,T Q ·|0,T . To do this, we define the loss function L(θ) which yields M θ = proj M (Π 2n ) at optimality: L DDSBM (θ) := KL(Π 2n ∥M θ ) = Z T 0 E Π 2n t,T X y̸=X t Q Q ·|T t log Q Q ·|T t Q M θ t (X t ,y) + (Q Q ·|T t −Q M θ t )(X t ,X t ) dt (7.80) where Π 2n =M φ 0,T Q ·|0,T . To sample from Π 2n , we first sample endpoints by simulating the reverse- time Markov generatorM φ to obtain (x 0 ,x T )∼ M φ 0,T and then sampling the reference bridge X t ∼ Q(·|x 0 ,x T ) For each intermediate sampleX t ∼ Q(·|x 0 ,x T ), we condition the forward-time reference generator to X T = x T defined in Lemma 7.16 as: Q Q ·|T t (X t ,y) = E X T ∼Π T|0,t Q 0 t (X t ,y;X T )|X t ,X 0 (7.81) Step 1b: Forward-Time Reciprocal Projection. Then, we define the corresponding recip- rocal projection Π 2n+1 =proj R(Q) (M 2n+1 ) as the mixture of bridges under the reference measure conditioned on the endpoint law of M θ defined as: Π 2n+1 = M θ 0,T Q ·|0,T (7.82) While we show that the Markovian projection preserves the bridge measureM t = Π t in theory (Proposition 4.10), in practice, parameterizing the generator of the forward CTMC with a neural networkθcan result in mismatches in the terminal marginal constraint at timet=T. Therefore, similarly to the DSBM algorithm in Section 6.3, we also parameterize the reverse-time Markovian projection that explicitly constrains the terminal marginal. Step 2a: Reverse-Time Markovian Projection. To obtain the reverse-time Markovian projection M 2n+2 =proj M (Π 2n+1 ) parameterized by the reverse generatorφ, we minimize the KL divergence KL(Π 2n+1 ∥M φ ) with reciprocal measure from the previous iteration Π 2n+1 =M θ 0,T Q ·|0,T and define 205 the loss function L(φ) which yields M φ = proj M (Π 2n+1 ) at optimality: L DDSBM (φ) := KL(Π 2n+1 ∥M φ ) = Z T 0 E Π 2n+1 0,t X y̸=X t ̃ Q Q ·|0 t log ̃ Q Q ·|0 t ̃ Q M φ t (X t ,y) + ( ̃ Q Q ·|0 t − ̃ Q M φ t )(X t ,X t ) dt (7.83) where the expectation is over samples from Π 2n+1 =M θ 0,T Q ·|0,T . Each sample from Π 2n+1 is obtained by simulating the forward-time Markov generatorM θ to obtain (x 0 ,x T )∼ M θ 0,T and then sampling the reference bridge X t ∼ Q(·|x 0 ,x T ). For each intermediate sampleX t ∼ Q(·|x 0 ,x T ), we condition the reverse-time reference generator to X 0 = x 0 defined analogously to the forward-time generator as: ̃ Q Q ·|0 t ( ̃ X t ,y) = E X 0 ∼Π 0|t,T h Q 0 t ( ̃ X t ,y;X T )| ̃ X t ,X 0 i (7.84) Step 2b: Reverse-Time Reciprocal Projection. Then, we define the corresponding recip- rocal projection Π 2n+2 =proj R(Q) (M 2n+2 ) as the mixture of bridges under the reference measure conditioned on the endpoint law of M φ defined as: Π 2n+2 = M φ 0,T Q ·|0,T (7.85) Now that we understand how the discrete diffusion SBM algorithm is implemented in practice, it is natural to question whether it yields the same convergence guarantees as the IMF procedure. Since we already establish the intuition and formal proofs for the convergence of the IMF procedure in R d in Section 4.5, the discrete state space analog follows similar intuition, where the IMF sequence results in monotonically decreasing KL divergence with the optimal bridgeP ⋆ which converges to zero at the limit when the number of iterations goes to infinity. 32 Remark 7.20 (Comparison of Discrete SB-SOC and DDSBM). A key distinction between the discrete diffusion Schr ̈odinger bridge method (DDSBM) and the discrete stochastic optimal control (SOC) formulation lies in their dependence on target samples. The DDSBM algorithm initializes the reciprocal measure as Π 0 :=π 0,T Q ·|0,T , which requires access pairs from both marginals, and then samples fromπ 0 when optimizing the forward-time Markovian projection and π T when optimizing the reverse-time Markovian projection. In contrast, the discrete SB-SOC framework (Box 7.4) only requires computing the RND between the optimally-controlled path measure and the one generated from the current control, enabling matching of unknown target distributions, such as temperature-annealed or reward-tilted distributions. 7.7 Closing Remarks for Section 7 In this section, we introduced the theory of continuous-time Markov chains (CTMCs), which provide the natural analogue of stochastic differential equations for modeling stochastic processes in discrete state spacesX=1,...,d. In this setting, the role of the control drift is replaced by a generator or transition rate matrixQ u t , which governs the jump dynamics of the process. Using this representation, we defined the discrete Schr ̈odinger bridge problem, which seeks the controlled CTMC path measureP u with generatorQ u t that remains closest, in KL divergence, to a reference path measure Q with generator Q 0 t , while satisfying the marginal constraints p 0 = π 0 and p T = π T . 32 for a rigorous proof that explicitly uses the CTMC generators, see Theorem 3.3 in (Kim et al., 2024). 206 Building on this formulation, we introduced two frameworks for solving the discrete Schr ̈odinger bridge problem: stochastic optimal control (SOC) and iterative Markovian fitting (IMF). These approaches mirror their continuous-state counterparts, translating the ideas of controlled SDEs and path-space projections to the setting of jump processes governed by CTMC generators. Taken together, we have shown that the Schr ̈odinger bridge framework extends naturally from diffusion processes in continuous spaces to jump processes in discrete state spaces. With this theoretical foundation in place, we are now prepared to explore how all the ideas developed throughout this guide can be applied in practice to solve real data-driven and scientific problems. In the final section, we examine several applications of generative modeling with Schr ̈odinger bridges, illustrating how the theoretical tools developed throughout this guide can be used to construct practical generative models across a variety of domains. 207 8. Applications of Generative Modeling with Schr ̈odinger Bridges While Schr ̈odinger bridges can be seen as a unified framework that encompasses a large portion of generative modeling techniques, from denoising diffusion to score-based generative modeling to flow matching, there are several applications where Schr ̈odinger bridge frameworks are specialized to solve. In this section, we highlight three prominent applications where advances in Schr ̈odinger bridge–based generative modeling have led to principled and practically effective solutions, including data trans- lation (Section 8.1), single-cell modeling (Section 8.2), and sampling high-Boltzmann energy distributions (Section 8.3). Our goal is not to provide an exhaustive review of applications, but to illustrate how learning entropy-regularized stochastic paths between structured distributions can be applied to diverse scientific and data-driven problems. 8.1 Data Translation Prerequisite: Section 6.1 The goal of data translation is to transform samples from one unknown distribution to another. Specifically, we will focus on image-to-image translation, which aims to learn a mapping between pairs of images and can be applied to many tasks such as deblurring, inpainting, and image editing, as illustrated in Figure 22. In this setting, the source and target datasets correspond to two probability distributions defined over the space of images, and the goal is to learn a transformation that transports samples from the source distribution to the target distribution while preserving the underlying structure of the data. The Schr ̈odinger bridge framework provides a principled probabilistic formulation of this problem. Given empirical samples from the source distributionπ 0 and the target distributionπ T , the Schr ̈odinger bridge constructs the most likely stochastic process that connects the two distributions while remaining close to a reference diffusion. Rather than learning a deterministic mapping between individual samples, the bridge defines a stochastic transport process that transforms an image in the source distribution into a many possible images in the target distribution. In many image restoration tasks the degraded image already contains substantial structural information about the target image. Rather than generating from random noise, Schr ̈odinger bridges allow the generative process to start directly from the degraded image distribution and gradually transform it into the clean image distribution. Since the data translation task assumes paired samples, we can consider the form of the SB problem, where the clean distributionπ T =δ x is a dirac Delta function at the target data point (e.g., clean image), which corresponds to some distribution of initial samplesx T ∼ π T (·|x 0 ) (e.g. degraded or corrupted images). This produces a joint distribution of paired samples defined as: π(x 0 ,x T ) = π 0 (x 0 )π(x T |x 0 )(8.1) Given this joint distribution, we can define a form of the SB problem which samples pairs from the joint data distribution and learns a Gaussian interpolation between the pair (Liu et al., 2023b). Definition 8.1 (Data Translation with Schr ̈odinger Bridge). Given data pairs sampled from the joint distribution (x 0 ,x T )∼ π 0 (x 0 )π T (x T |x 0 ), the optimal Schr ̈odinger bridge density at intermediate times t∈ [0,T ] takes the Gaussian form: X t ∼ p t (X t |x 0 ,x T ) =N (X t ;μ t (x 0 ,x T ), Σ t )(8.2) whereμ t ∈ R d is the conditional mean at timet, Σ t ∈ R d×d is the covariance matrix. A choice of parameterization used in image restoration (Liu et al., 2023b) definesσ t ∈ Ras the variance accumulated in forward time, and ̄σ t ∈ Ras the variance accumulated in the reverse 208 Figure 22: Image to Image Translation with Schr ̈odinger Bridges. Schr ̈odinger bridges enable image-to-image translation by learning a stochastic process that transports samples between a structured prior distributionπ 0 (e.g., corrupted or partially observed images) and a clean target distributionπ T (right to left). Rather than generating images from pure Gaussian noise, the model learns the optimal stochastic interpolation conditioned on the endpoints, corresponding to the dynamic optimal transport map when diffusion is small. This yields generation trajectories that progressively restore structure, for tasks such as inpainting, deblurring, or translation, leading to more efficient and interpretable transformations than denoising from noise alone. time coordinate, which explicitly define the mean and covariance as: μ t = ̄σ 2 t ̄σ 2 t + σ t x 0 + σ 2 t ̄σ 2 t + σ t x T , Σ t = σ 2 t ̄σ 2 t ̄σ 2 t + σ 2 t I d , σ 2 t := Z t 0 β s ds, ̄σ 2 t := Z T t β s ds (8.3) Whenβ t ≡ βis constant overt∈[0,T] and sufficiently small, the optimal SB reduces to the dynamic optimal transport map between the pair (x 0 ,x T ) defined as: v t (X t ) = X t −x 0 t , μ t = 1− t T x 0 + t T x T (8.4) To learn the velocityv θ that optimally transports samples fromπ 0 toπ T , we optimize the matching loss defined as: L(θ) := E x 0 ∼π 0 (X 0 ),x T ∼π T (X T |X T ) Z T 0 v θ (X t ,t)− X t −x 0 σ t |z ∇ log ˆφ t (X t |x 0 ) 2 (8.5) where∇ log ˆφ t (X t |X 0 ) is the SB potential drift which is equivalent to the conditional score function ∇ logp t (X t |x 0 ) where X t ∼ p t (·|x 0 ,x T ). We observe that this objective resembles the standard conditional score matching objective, where the score function is given by∇ logp t (X t |y), whereyis the conditioning information, which in the case of data translation is the initial or corrupted sample. However, in conditional score matching, the generation process starts from Gaussian noisex 0 ∼ N(0,I d ) and learns the score function using the corrupted sample as just another input to the model. When the corrupted sample is already close to the target distribution, generating the clean target from pure noise is highly inefficient, which reveals the advantages of Schr ̈odinger bridges as a way to 209 Figure 23: Modeling Cellular Dynamics with Schr ̈odinger Bridges. Cellular processes such as differentiation, perturbation responses, and development can be viewed as stochastic trajectories through a high-dimensional cell state space inferred from single-cell measurements. Because single-cell sequencing provides only static snapshots of cell populations at discrete times, the underlying dynamics are not directly observed. Schr ̈odinger bridges infer the most likely stochastic process that transports an initial cell distributionπ 0 to a later observed distributionπ T , yielding a probabilistic model of cell state evolution that captures both deterministic differentiation trends and stochastic biological variability. learn the optimal interpolating bridge between a structurally informative priorπ 0 to the target distributionπ T . Since the SB dynamics explicitly connect corrupted and clean image distributions, the generative trajectories correspond to progressive restoration processes rather than denoising from pure noise, leading to more interpretable and efficient generation. 8.2 Modeling Single-Cell State Dynamics Prerequisites: Sections 5.2, 5.3, 5.4, 5.5 Cellular systems undergo dynamic state transitions from cell differentiation to responses under genetic or drug perturbations to adaptive changes during development and disease. These processes can be viewed as trajectories through a high-dimensional cell state spaceR d , typically defined as single-cell RNA sequencing (scRNA-seq) measurements of gene expression. Due to the destructive nature of single-cell sequencing technologies, which kill the cell after mea- surement, data is often a collection of static snapshots of populations at discrete time points rather than continuous-time measurements. As a result, the underlying stochastic dynamics governing how cells transition between states are not directly observed. Instead, the problem becomes one of inferring the dynamical process that transports a distribution of cells at an initial condition π 0 ∈P(R d ) to a distribution observed at a later time π T ∈P(R d ). π 0 (x) |z initial cell distribution differentiation, perturbation, etc. −→π T (x) |z terminal cell distribution , x∈ R d (8.6) Since cell states naturally lie on a non-linear biological manifold, the Schr ̈odinger bridge framework provides a natural approach to this problem. Given empirical distributions of cellular states at sequential time pointsπ t k K k=1 , the Schr ̈odinger bridge identifies the most likely stochastic process that connects these distributions while remaining close to a chosen reference diffusion. In this formulation, the inferred bridge represents a probabilistic model of cell state evolution that captures both the deterministic drift driving differentiation and the stochastic variability inherent in biological systems. 210 Modeling Cellular Dynamics with Schr ̈odinger Bridges Consider a sequence of snapshots of a cell population at discrete time intervalsπ t k ∈P(R d ) K k=1 for 0 =t 0 <· < t k <· < t K =TwhereR d represents ad-dimensional gene expression space. Then, the problem of modeling cellular dynamics between snapshots can framed as optimizing a parameterized controlu θ (x,t) that minimizes a combination of the (Multi-Marginal SB Problem) and the (Generalized SB Problem): inf u θ E X 0,T ∼P u Z T 0 1 2 ∥u θ (X,t)∥ 2 +I(X,p t ,t) dt (Generalized SB Problem) s.t. ( dX t = (f (x,t) + σ t u θ (X t ,t))dt + σ t dB t X 0 ∼ π 0 , X T ∼ π T (8.7) wheref(x,t) is the underlying drift of the cell population,σ t is the diffusion coefficient representing stochastic in cell evolution, andI(x,p t ) :R d ×P(R d )→ Ris the interaction cost that measures how the cells evolve with respect to the overall population. When the population of cells changes over time, we can formulate the cell modeling problem as a (Unbalanced SB Problem) with unbalanced marginal constraints: inf p t ,u θ ,g φ Z T 0 Z R d 1 2 ∥u θ (x,t)∥ 2 + αΨ(g φ (x,t)) p t (x)dxdt(8.8) s.t. ( ∂ t p t =−∇· (p t (f + σ t u) + σ 2 t 2 ∆p t + gp t ∀k ∈1,...,K, p t k = π t k whereg φ (x,t) parametrizes the growth rate that determines how the cell population increase or decrease in mass and Ψ(g φ (x,t)) is the scalar function that penalizes cell growth and death. Finally, we can consider the case of branching cell dynamics, where the terminal cell population contains multiple distinct modesπ T =π T,k K k=1 defining different sub-populations that have diverged as a result of differentiation or cellular perterbation. This problem can naturally be framed as the (Branched SB Problem), which can be solved by defining a set of parameterized control drifts and growth rates u θ,k ,g φ,k K k=1 that minimize: inf u θ,k ,g φ,k K k=1 Z T 0 E p t,0 1 2 ∥u θ,0 (X t,0 ,t)∥ 2 + c(X t,0 ,t) w t,0 + K X k=1 E p t,k 1 2 ∥u θ,k (X t,k ,t)∥ 2 + c(X t,k ,t) w t,k dt(8.9) s.t. dX t,k = (f (X t,k ,t) + σ t u θ,k (X t,k ,t))dt + σ t dB t X 0 ∼ π 0 , X T,k ∼ π T,k w 0,k = δ k=0 , w T,k = w ⋆ T,k where the weight of the primary branch is given byw t,0 = 1 + R t 0 g t,φ (X s,0 ,s)dsand the weights of the K secondary branches is given by w t,k = R t 0 g k,φ (X s,k ,s)ds. This shows that Schr ̈odinger bridges defines a unified problem that can be specialized to several biologically relevant settings. In the balanced setting, the bridge captures smooth state transitions between multiple cell population distributions. In the unbalanced setting, it incorporates cell proliferation and death through growth terms, allowing the total population mass to vary over time, and finally, the branched Schr ̈odinger bridge formulation models lineage diversification and 211 diverse responses to perturbations by decomposing the population into multiple sub-trajectories, each governed by its own dynamics and weights. Together, these formulations provide a principled and flexible framework for modeling complex cellular processes, including differentiation, perturbation responses, and lineage branching (Tang et al., 2026; Zhang et al., 2024; 2025b). By learning the underlying bridge dynamics, we not only interpolate intermediate cellular states but also recover the latent structure of population evolution, enabling predictive modeling of how cell populations evolve under changing biological conditions. 8.3 Sampling Boltzmann Distributions The sampling problem aims to generate a complete reconstruction of a probability distribution. A specific class of distributions of particular interest in computational sciences are Boltzmann distributions, which are unnormalized and often high-dimensional and defined only by an energy functionU(x) :R d → Rwithout explicit samples (Binder et al., 1992; Tuckerman, 2023; Yang et al., 2019). Definition 8.2 (Sampling Boltzmann Distributions). Given an energy functionU(x) :R d → R where R d , the Boltzmann distribution is defined as: π T (x) := e −U (x) Z ,where Z := Z R d e −U (x) dx(8.10) such that R R d π T (x)dx = 1 and Z is some, often intractable, normalization factor. This class of distributions are particularly difficult to sample due to the computational cost of evaluating the energy function of the molecular systems. Traditional approaches such as Markov- chain Monte Carlo (MCMC) (Metropolis et al., 1953; Neal, 2001) and Sequential Monte Carlo (SMC) (Doucet et al., 2001) while provably converges to the target distributionπ T , exhibit slow mixing times, where the sampled distribution requires a significant number of simulation steps to converge to the target distribution, and can remain trapped in local minima with poor reconstruction of the global energy landscape, especially for large molecules with high-dimensional representations and multi-modal energy landscapes. To overcome these challenges, generative modeling frameworks have been developed that approach the problem as transporting a easy-to-sample source distributionX 0 ∼ π 0 (e.g., a Gaussian or Dirac delta at the origin) to the target Boltzmann distributionπ T (x) = e −U(x) Z . Speciically, these frameworks define parameterized control driftu θ (x,t) that steers the sample toward the target Boltzmann distribution π T (x) = e −U(x) Z through the controlled SDE: dX t = [f t (X t ) + σ t u θ (X t ,t)]dt + σ t dB t , X 0 ∼ π 0 (8.11) In this setting, we do not have access to explicit samples from the target distributionπ T and can only evaluate its unnormalized density through an energy functionπ T (x) = e −U(x) Z . This implicit specification makes direct sampling intractable, and naturally motivates a formulation using stochastic optimal control (SOC) as described in Section 3, where we optimize the control u θ (x,t) such that the terminal distribution generated from the optimal controlu ⋆ (x,t) matches the Boltzmann distribution p ⋆ T = π T . Specifically, we consider the (SB-SOC Objective) described in Section 3.2, where the terminal constraint is defined aslog ˆφ T (X T ) π T (X T ) , whereˆφ T is the backward Schr ̈odinger potential evaluated at 212 Figure 24: Sampling the Boltzmann Distribution with Schr ̈odinger Bridges. Boltzmann distributions π T (x)∝ e −U (x) arise in many scientific applications but are often difficult to sample from due to the intractable normalization constantZ. Schr ̈odinger bridges address this by learning a controlled stochastic process that transports samples from an easy-to-sample priorπ 0 (e.g., a Gaussian) to the target Boltzmann distribution. The learned control drift guides trajectories so that their terminal distribution matchesπ T , enabling efficient sampling from energy-based models. time t = T . When sampling from a Boltzmann distribution, this constraint becomes: log ˆφ T (X T ) π T (X T ) = log ˆφ T (X T )− log e −U (X T ) Z = log ˆφ T (X T ) + U (X T ) + logZ(8.12) Using this definition for the terminal constraint, we can define the Boltzmann SB-SOC problem. Proposition 8.3 (Sampling the Boltzmann Distribution with Schr ̈odinger Bridges and Stochastic Optimal Control ((Liu et al., 2025))). The optimal control driftu ⋆ (x,t) that generates samples from the Boltzmann distributionπ T (X T ) = e −U(X T ) Z solves the stochastic optimal control (SOC) problem defined as: inf u E X 0:T ∼P u θ " Z T 0 1 2 ∥u θ (X t ,t)∥ 2 dt + log ˆφ T (X T ) + U (X T ) + logZ # (8.13) s.t. dX t = (f (X t ,t) + σ t u θ (X t ,t))dt + σ t dB t , X 0 ∼ π 0 whereˆφ T (x) is the backward Schr ̈odinger potential. Under the optimal controlu ⋆ (x,t), the generated distribution p ⋆ T exactly reconstructs the Boltzmann distribution p ⋆ T = π T . Proof. This result follows immediately from our proof of Proposition 3.11 by replacing the terminal value constraint with log ˆφ T (X T ) π T (X T ) = log ˆφ T (X T ) + U (X T ) + logZ to get: P ⋆ (X 0 ,X T ) = 1 Z Q(X 0 ,X T ) exp (− log ˆφ T (X T )− U (X T )− logZ− logφ 0 (X 0 ))(8.14) 213 and integrating overX T to get the terminal distribution generated from the optimal control driftu ⋆ : p ⋆ T (X T ) = exp (−U (X T )− logZ) = π T (X T )(8.15) which exactly matches the target Boltzmann density without the initial value function bias. □ One approach to solving this problem is using the (SB-AM Objective) and (Corrector Matching Objective) (Liu et al., 2025), defined speciically for sampling Boltzmann density as: L SB-AM (u) := E p ̄u 0,T " 1 2 Z T 0 E p ̄u t|0,T h ∥u(X t ,t) + σ t (∇U (X T ) +∇ log ˆφ T (X T ))∥ 2 i dt # (8.16) L SB-CM ( b Z T ) := E p ̄u 0,T h ∥ b Z T (X T )−∇ x T log Q T|0 (X T |X 0 )∥ 2 i (8.17) Since the prior distribution for the sampling problem is typically Gaussian and the intermediate dynamics can be arbitrarily defined as long as it converges to the target distribution, we consider the case whereQis pure Brownian motion with zero driftf t := 0 and the conditional distributionp ̄u t|0,T takes a tractable form without requiring sampling the full path 33 X 0:T ∼ P ̄u . This perspective reveals that sampling from Boltzmann distributions can be interpreted as learning the optimal Schr ̈odinger bridge between a simple prior distribution and the target energy-based distribution. We show that formulation yields efficient algorithms that significantly reduce the number of energy evaluations and enables fast inference by simulating an SDE with a parameterized control drift. Notably, these examples highlight the broader potential of Schr ̈odinger bridge methods as a scalable framework for sampling in high-dimensional and complex energy landscapes, with promising applications in modeling and understanding diverse molecular properties. 8.4 Closing Remarks for Section 8 This section illustrates notable examples of how Schr ̈odinger bridge formulations extend beyond a unifying theoretical framework to become powerful, task-specific tools for generative modeling. Across data translation, single-cell dynamics, and sampling from complex energy landscapes, the common principle is the learning of entropy-regularized stochastic paths that respect both the structure of the data and the underlying physical or statistical constraints. By framing these problems as controlled stochastic processes, Schr ̈odinger bridges provide a flexible mechanism for incorporating domain knowledge, handling distributional mismatch, and enabling efficient inference in high-dimensional settings. These successes highlight the versatility of the framework and suggest that Schr ̈odinger bridge–based methods will continue to play a central role in advancing generative modeling across scientific and data-driven domains. 33 See (Liu et al., 2025) Appendix A for explicit definition 214 9. Conclusion This guide aims to serve as a self-contained theoretical exploration of the foundations of Schr ̈odinger bridges and their role in modern generative modeling. By starting from fundamental concepts in optimal transport, stochastic processes, and information theory, we progressively build toward the theory of constructing Schr ̈odinger bridge and modern generative modeling algorithms leveraging Schr ̈odinger bridge theory to model real-world, high-dimensional systems and data. Throughout this guide, we illustrate how the Schr ̈odinger bridge problem provides a unifying perspective on entropy- regularized optimal transport, stochastic optimal control, and score-based generation. Rather than viewing generative modeling as a collection of disconnected algorithmic techniques, the Schr ̈odinger bridge framework reveals a unified mathematical structure underlying these methods. Building on top of this foundation exist extensions to diverse problem settings and algorithmic innovations. One of the central themes of this guide is that generative modeling can be understood as a problem of probability transport in the space of path measures. In this view, the goal is not just to map one distribution into another through a deterministic map, but to identify the most likely stochastic process that forms a continuous-time bridge between distributions them under a given reference process. This perspective unifies several seemingly distinct paradigms—including score-based generative models, flow matching, and stochastic optimal control—under a common variational principle based on relative entropy between path measures. Looking ahead, Schr ̈odinger bridge methods are likely to play an increasingly important role in the development of generative modeling and probabilistic machine learning. Ongoing work on scalable bridge solvers (Gushchin et al., 2024; Zhang et al., 2025a), stochastic control formulations (Liu et al., 2025; Tang et al., 2025a), application-focused algorithms (Li et al., 2025a; Li et al., 2025b), and extensions to the discrete state space (Guo et al., 2026; Ksenofontov et al., 2025) suggests the adaptibility and scalability of this framework. Notably, emerging applications of Schr ̈odinger bridges for scientific applications—ranging from prediction of drug perturbations on cell populations (Tang et al., 2026) and cell differentiation modeling (Zhang et al., 2024) to molecular dynamics simulation (Du et al., 2024; Tang et al., 2025a) and sampling high-dimensional molecular energy landscapes (Liu et al., 2025)—highlight the potential of bridge-based methods for modeling complex stochastic processes. We hope that this guide serves both as a conceptual foundation and as a practical reference for researchers, practitioners, and learners interested in the theory and applications of Schr ̈odinger bridges. By building the mathematical principles that underpin modern generative models, we aim to provide a framework for understanding existing methods and for developing new approaches that bridge the gap between theory and application-focused design. Note from the Author I deeply thank all of my research mentors and collaborators for all the educational and inspiring conversations throughout my time in this field. 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