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On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups
Hamza Mahmood, Georgiy A. Bondar, Abhishek Halder, Adeel Akhtar
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Summary
This paper establishes the equivalence between the stochastic optimal control formulation and the path space formulation of the Schrödinger bridge problem (SBP) on compact connected Lie groups. By utilizing horizontal lift and stochastic anti-development, the authors derive a Girsanov-type change-of-measure result, proving that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. This equivalence allows the SBP to be viewed as a relative entropy minimization problem, facilitating proofs of existence and uniqueness, providing a large-deviation probabilistic interpretation, and enabling computational solutions via static Sinkhorn recursions.
Entities (10)
Relation Signals (8)
Schrödinger bridge problem → hasformulation → Stochastic Optimal Control
confidence 95% · We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP)
Schrödinger bridge problem → hasformulation → Path space formulation
confidence 95% · We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP)
Stochastic Optimal Control → isequivalentto → Path space formulation
confidence 95% · We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP)
Expected control energy → equals → Relative Entropy
confidence 93% · show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure.
Schrödinger bridge problem → appliedon → Compact connected Lie group
confidence 92% · We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group.
Relative entropy minimization → solvedby → Sinkhorn recursions
confidence 90% · From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem
Stochastic anti-development → usedin →
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Abstract
Abstract:We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint this http URL result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: this https URL
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- Source: https://arxiv.org/abs/2609.13758v1
- Canonical: https://arxiv.org/abs/2609.13758v1
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On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups Thanks: This research has been supported by NJIT’s startup funds.Thanks: H. Mahmood∗ and A. Akhtar∗ are with the Dept. of Mechanical & Industrial Engineering at the New Jersey Institute of Technology, Newark, NJ 07102, USA (email: hm576, adeel.akhtar@njit.edu). G.A. Bondar† is with the Department of Applied Mathematics at the University of California at Santa Cruz, CA, 95064, USA. (email: gbondar@ucsc.edu) A. Halder‡ is with the Dept. of Aerospace Engineering at Iowa State University, Ames, IA 50011, USA (email: ahalder@iastate.edu). Hamza Mahmood* Georgiy A. Bondar Abhishek Halder Adeel Akhtar* Abstract We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals. Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus 2T^2. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP I Introduction The Schrödinger bridge problem (SBP) concerns steering a stochastic system between prescribed endpoint probability distributions while minimizing the expected control effort [1, 2]. A substantial systems-control literature has developed around this problem, including theoretical [3, 4, 5, 6, 7] and computational/application-oriented works [8, 9], primarily for Euclidean state spaces. In a recent work [10], we developed a coordinate-free formulation of the SBP on a compact connected Lie group, established existence-uniqueness of solution through the associated Schrödinger system, and obtained a dynamic Sinkhorn scheme for computing the optimal bridge. However, that work did not address the corresponding path space formulation or its large-deviation interpretation, which identifies the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the prescribed endpoint constraints. A related path space formulation for an SBP on Euclidean space was given in [9, Theorem 4.2] using Girsanov’s theorem. For Lie group valued Brownian motion, a Girsanov type result was developed in [11] for matrix Lie groups by transforming a multiplicative Girsanov problem to the Lie algebra through logarithm and exponential mappings. In contrast, the diffusion considered here evolves through tangent spaces that vary along the trajectory. We use horizontal lift and stochastic anti-development [12, Ch. 2] to represent this diffusion in Euclidean coordinates and carry out the required change of measure intrinsically on the Lie group. We make the following contributions. 1. We derive a Girsanov type change-of-measure result for the controlled diffusion on the Lie group using horizontal lift and stochastic anti-development (Lemma 1). 2. We establish the equivalence between expected control energy and the relative entropy of the controlled path law with respect to the reference Wiener measure (Corollary 1), thereby recasting the stochastic control version of the SBP as a path space relative entropy minimization problem. 3. We show three implications of this equivalence: analytically, it yields existence-uniqueness of the SB (Theorem 1); probabilistically, it provides a large-deviation interpretation of the SB as the most probable deviation of the uncontrolled dynamics consistent with the endpoint constraints; numerically, it enables computation of the optimal path measure via static Sinkhorn recursions. We numerically illustrate the equivalence on the torus 2T^2. Notations. For a Lie group G and R∈R∈ G, the notation R T_R G denotes the tangent space of G at R. We use C([0,1],)C([0,1]; G) to denote the space of continuous paths from [0,1][0,1] into G. The symbols Δ _ G and divdiv denote, respectively, the Laplace–Beltrami operator on G and the divergence operator. In a stochastic differential equation (SDE), the symbol ∘ indicates that the stochastic integral is understood in the Stratonovich sense. For two measures ℙP and ℚQ on some measure space X, the notation ℙ≪ℚP means that ℙP is absolutely continuous with respect to ℚQ. The relative entropy or Kullback-Leibler (KL) divergence DKL(ℙ∥ℚ):=ℙ[logdℙdℚ]=∫log(dℙdℚ)dℙ,if ℙ≪ℚ,D_KL(P ):=E_P [ dPdQ ]= _X ( dPdQ )dP,\;if P , and +∞+∞ otherwise, where ℙ[⋅]E_P [· ] denotes the expectation operator w.r.t. the measure ℙP, and dℙdℚ dPdQ denotes the Radon-Nikodym derivative of ℙP w.r.t. ℚQ. The symbol ∥⋅∥2\|·\|_2 denotes the standard Euclidean norm, whereas ∫0t⟨αs,dβs⟩:=∫0t∑i=1nαsidβsi _0^t _s,d _s := _0^t _i=1^n _s^i\,d _s^i denotes the Euclidean inner product in stochastic integral. For a topological space A, its Borel σ-algebra is denoted by ℬ(A)B(A). For two σ-algebras ℱ1,ℱ2F_1,F_2, their product σ-algebra ℱ1⊗ℱ2F_1 _2 is generated by sets A1×A2A_1× A_2 with Ai∈ℱiA_i _i, i∈1,2i∈\1,2\. A filtration ℱtt∈[0,1]\F_t\_t∈[0,1] on a probability space (Θ,ℱ,ℙ)( ,F,P) is an increasing family of sub-σ-algebras of ℱF. Other notations will be introduced in situ. I Equivalence of the Stochastic Control and Path Space Formulations This section formulates the SBP as a stochastic optimal control problem, develops the path space change-of-measure construction on the Lie group, and establishes the equivalence between the stochastic control and path space relative entropy formulations. I-A Background Let G be a compact connected Lie group with identity element e, and let =e g= T_e G be its finite-dimensional Lie algebra of dimension n. Consider an inner product ⟨⋅,⋅⟩:×→ℝ ·,· _ g: g× g and an orthonormal basis e1,e2,…,en\e_1 g,e_2 g,…,e_n g\ of g w.r.t. ⟨⋅,⋅⟩ ·,· _ g . Let ⋅^:ℝn→ ·:R^n→ g, called the “hat map”, denote an isomorphism between the vector spaces ℝnR^n and g. For any R∈R∈ G, we define the left multiplication LR:→L_R: G→ G by LR(g)=RgL_R(g)=Rg. Its differential at the identity, (LR)e:e→R,( DL_R)_e: T_e G→ T_R G, transports elements of g to tangent vectors at R. The controlled kinematic equation [13, Ch. 5] on G is the following (deterministic) differential equation: R˙t=(LRt)e(Ω^(Rt,t)),R˙t:=dtRt, R_t=( DL_R_t)_e ( (R_t,t) ), R_t:= ddtR_t, (1) where Rt∈R_t∈ G denotes the configuration (i.e., state) of the system evolving on G at time t, with its time derivative R˙t∈Rt R_t∈ T_R_t G, and the control input Ω:×[0,1]→ℝn : G×[0,1] ^n. The controlled stochastic kinematic equation is a noisy generalization of (1) that describes a diffusion process R(t)R(t) on G. This process solves the controlled Stratonovich SDE: dRt=(LR)eΩ^(R,t)dt+σ∑i=1n(LR)eei∘dWti.dR_t=( DL_R)_e (R,t)\>dt+σ _i=1^n( DL_R)_e\>e_i g ^i_t. (2) In (2), WtW_t is an ℝnR^n-valued standard Brownian motion, and σ>0σ>0 is the (isotropic) diffusion strength. I-B The Schrödinger Bridge Problem Let μ be the Haar (volume) measure on G and let 2()P_2( G) denote the set of PDFs on G with finite second moments, i.e., 2():=ρ:→ℝ≥0|∫ρdμ(R)=1, _2( G) := \ρ: G _≥ 0 | _ Gρ\,dμ(R)=1, ∫dist(R,R0)2ρdμ(R)<∞, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; _ Gdist(R,R_0)^2\,ρ\,dμ(R)<∞ \, (3) where dist(R,R0)dist(R,R_0) denotes the Riemannian distance of R∈R∈ G from a fixed R0∈R_0∈ G. Given ρ0,ρ1∈2() _0, _1 _2( G), let 01()P_01( G) be the set of all probability density trajectories ρ(⋅,t)ρ(·,t) in 2()P_2( G) that are continuous in t∈[0,1]t∈[0,1] with endpoints ρ(⋅,0)=ρ0ρ(·,0)= _0 and ρ(⋅,1)=ρ1ρ(·,1)= _1, i.e., 01():=ρ:×[0,1]→ℝ≥0|ρ(⋅,0)=ρ0,ρ(⋅,1)=ρ1, _01( G) := \ρ: G×[0,1] _≥ 0\, |\,ρ(·,0)= _0,ρ(·,1)= _1, for each t∈[0,1],ρ(⋅,t)∈2(). \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,for each t∈[0,1],ρ(·,t) _2( G) \. (4) Also, let V be the set of all Markovian control policies on G with finite energy, i.e., :=Ω:×[0,1]→ℝn|Ω is measurable and := \ : G×[0,1] ^n\; |\; is measurable and (5) ∀ρ∈01(),∫01∫∥Ω(R,t)∥22ρ(R,t)dμ(R)dt<∞. ∀ρ _01( G), _0^1 _ G\| (R,t)\|_2^2\,ρ(R,t)\,dμ(R)\>dt<∞ \. The main idea in SBP is to design a control Ω∈ that steers the stochastic state R on G from a prescribed PDF ρ0 _0 to another ρ1 _1 over a finite time horizon [0,1][0,1] with minimum expected control effort. The controlled stochastic process R is governed by the stochastic differential equation (SDE) (2) together with R0∼ρ0(R),R1∼ρ1(R).R_0 _0(R),\;R_1 _1(R). Then, the corresponding controlled state PDF ρ(R,t)ρ(R,t) must satisfy the constraints [10, equation (6b)]: ∂tρ=−div(ρ(LR)eΩ^(R,t))+σ22Δρ(R,t), _tρ=-div(ρ( DL_R)_e (R,t))+ σ^22 _ Gρ(R,t), (6a) ρ(R,0)=ρ0(R),ρ(R,1)=ρ1(R). ρ(R,0)= _0(R), ρ(R,1)= _1(R). (6b) We can now formally state the SBP for the kinematic equation on G as follows. Problem 1. Solve stochastic optimal control problem: inf(ρ,Ω)∈01()×∫01∫12‖Ω(R,t)‖22ρ(R,t)μ(R)t _ (ρ, ) _01( G)×V\; _0^1 _ G 12\| (R,t)\|_2^2\,ρ(R,t)dμ(R)dt (7a) subjecttoconstraints(6). \,to\,\,\,\,\,\,\,\,\,constraints~ eq:FPK_PDE. (7b) Problem 1 therefore seeks a finite-energy Markovian control that steers the stochastic process on G between the prescribed endpoint distributions while minimizing the expected control energy. The associated density evolution ρ∈01()ρ _01( G) satisfying (6) generates a stochastic interpolant between these marginals. We next develop the path space formulation and establish its equivalence with the control formulation (7). I-C Path Space Setup and Stochastic Anti-Development We recall that a stochastic process Y=(Yt)t∈[0,1]Y=(Y_t)_t∈[0,1] with values in a measurable space (E,ℰ)(E,E) is said to be progressively measurable [14, Definition 1.11] with respect to a filtration ℱtt∈[0,1]\F_t\_t∈[0,1] if, for every t∈[0,1]t∈[0,1], the mapping (s,θ)⟼Ys(θ),(s,θ)∈[0,t]×Θ,(s,θ) Y_s(θ),\;\;(s,θ)∈[0,t]× , is measurable from ([0,t]×Θ,ℬ([0,t])⊗ℱt) ([0,t]× ,B([0,t]) _t ) to (E,ℰ)(E,E). When the filtration is the usual augmentation of the canonical filtration with respect to a probability measure ℙP, we simply say that Y is progressively measurable with respect to the canonical filtration under ℙP. In particular, the Assumption 1 below requires that the process t↦Ω(Rt,t)t (R_t,t) be progressively measurable in this sense. An SDE is said to be well posed in law for a given initial distribution if a weak solution [14, Definition 3.1] exists and its probability law is unique: that is, if (R,W)(R,W) and (R~,W~)( R, W) are any two weak solutions, possibly defined on different filtered probability spaces but having the same prescribed initial distribution, then R and R~ R induce the same probability measure on the path space C([0,1],)C([0,1]; G). Let XiX_i be the left-invariant vector field on G associated with eie_i g, given by Xi(R):=(LR)eei,i=1,…,n.X_i(R):=( DL_R)_ee_i g, i=1,…,n. For a=(a1,…,an)⊤∈ℝna=(a_1,…,a_n) ^n, set Ra:=∑i=1naiXi(R). E_Ra:= _i=1^na_iX_i(R). This map R:ℝn→R E_R:R^n→ T_R G is an isometry. Indeed, the tangent space R T_R G is endowed with the inner product induced by the left-invariant Riemannian metric, namely ⟨v,w⟩R=⟨(LR−1)Rv,(LR−1)Rw⟩. v,w _R= ( DL_R^-1)_Rv,\,( DL_R^-1)_Rw _ g. Since eii=1n\e_i g\_i=1^n is orthonormal in g, it follows that R:ℝn→R E_R:R^n→ T_R G preserves inner products and hence is an isometry. Let :=C([0,1],),X:=C([0,1]; G), equipped with its Borel σ-algebra and canonical filtration, and let ρ0σW_ _0^σ be the Wiener measure on X associated with the reference diffusion dRt=σ∑i=1nXi(Rt)∘dWti,R0∼ρ0,dR_t=σ _i=1^nX_i(R_t) _t^i, R_0 _0, (8) where (Wt)t≥0(W_t)_t≥ 0 is an ℝnR^n-valued standard Brownian motion and ρ0∈2() _0 _2( G) given. Let ∇ denote the Levi-Civita connection associated with the left-invariant Riemannian metric on G induced by ⟨⋅,⋅⟩ ·,· _ g [15, Section 21.3]. Let Ut:ℝn→RtU_t:R^n→ T_R_t G denote the horizontal lift [12, Chapter 2] of the canonical process RtR_t with respect to ∇, initialized by U0=R0U_0= E_R_0, and let βt:=1σ∫0tUs−1∘dRs _t:= 1σ _0^tU_s^-1 _s (9) denote its stochastic anti-development. Under ρ0σW_ _0^σ, β is a standard ℝnR^n-valued Brownian motion. Next we define the map t:ℝn→ℝn O_t:R^n ^n as the composition of Ut−1U_t^-1 and Rt E_R_t: namely t:=Ut−1Rt. O_t:=U_t^-1 E_R_t. This map t O_t is an orthogonal transformation of ℝnR^n. For a Markovian control Ω∈ , we let bΩ(R,t):=RΩ(R,t)=(LR)eΩ^(R,t),b (R,t):= E_R (R,t)=( DL_R)_e (R,t), (10) and αt:=1σtΩ(Rt,t). _t:= 1σ O_t (R_t,t). (11) We now state the assumptions needed for our setup. Assumption 1 (Progressive measurability). The process t⟼Ω(Rt,t)t (R_t,t) is progressively measurable with respect to the canonical filtration under ρ0σW_ _0^σ. Assumption 2 (Well-posedness in law). The controlled Stratonovich SDE dRt=bΩ(Rt,t)dt+σ∑i=1nXi(Rt)∘dWti,R0∼ρ0,dR_t=b (R_t,t)\,dt+σ _i=1^nX_i(R_t) _t^i, R_0 _0, (12) is well posed in law. Assumption 3 (Novikov condition). The control Ω∈ satisfies ρ0σ[exp(12σ2∫01‖Ω(Rt,t)‖22t)]<∞.E_W_ _0^σ [ ( 12σ^2 _0^1\| (R_t,t)\|_2^2\,dt ) ]<∞. (13) Remark 1. The Assumptions 1-3 ensure the Girsanov change-of-measure construction on G . In particular, progressive measurability guarantees that the control can be used as an admissible integrand in the stochastic exponential, while well-posedness in law ensures that the process obtained after the change of measure can be identified uniquely with the controlled diffusion. The Novikov condition, in turn, guarantees that the corresponding stochastic exponential is a martingale and hence defines a probability measure on the path space. These standard assumptions provide natural sufficient conditions for the change-of-measure argument, and are mild enough to encompass a broad class of controlled diffusions relevant to our setting. I-D Girsanov Type Change of Measure Under Assumptions 1– 3, we can relate the controlled and uncontrolled path measures through an explicit Radon-Nikodym derivative, which will be essential for the relative entropy formulation of the SBP. Compared with the Euclidean case, the main additional difficulty on G is that the driving noise takes values in varying tangent spaces rather than in a fixed vector space. To apply the classical Girsanov theorem [16, Theorem 8.6.6], we will therefore use stochastic anti-development to represent the diffusion in Euclidean coordinates, and relate the horizontal frame to the left-invariant frame through the orthogonal map t O_t. This idea is used in the following lemma. Lemma 1 (Girsanov type result for a controlled diffusion on G). Under Assumptions 1– 3, the stochastic exponential Zt:=exp(∫0t⟨αs,dβs⟩−12∫0t‖αs‖22s)Z_t:= ( _0^t _s,d _s - 12 _0^t\| _s\|_2^2\,ds ) (14) is a strictly positive ρ0σW_ _0^σ-martingale with ρ0σ[Zt]=1E_W_ _0^σ[Z_t]=1. Define the probability measure ℙΩP on X by dℙΩdρ0σ=Z1. dP dW_ _0^σ=Z_1. Then βtΩ:=βt−∫0tαss _t := _t- _0^t _s\,ds (15) is a standard ℝnR^n-valued Brownian motion under ℙΩP . Furthermore, under ℙΩP , the canonical process R has the law of the controlled diffusion (12), and dℙΩdρ0σ dP dW_ _0^σ =exp(1σ∫01⟨tΩ(Rt,t),dβt⟩CLOSE = ( 1σ _0^1 O_t (R_t,t),d _t −12σ2∫01∥Ω(Rt,t)∥22dt). - 12σ^2 _0^1\| (R_t,t)\|_2^2\,dt ). (16) Proof. By the Novikov condition (Assumption 3), Zt=exp(∫0t⟨αs,dβs⟩−12∫0t‖αs‖22s)Z_t= ( _0^t _s,d _s - 12 _0^t\| _s\|_2^2\,ds ) is a strictly positive martingale satisfying ρ0σ[Zt]=1.E_W_ _0^σ[Z_t]=1. We can therefore define the probability measure ℙΩP on X by dℙΩ=Z1dρ0σ.dP =Z_1\,dW_ _0^σ. Since Z1>0Z_1>0, ρ0σW_ _0^σ-almost surely, the two measures are mutually absolutely continuous: ℙΩ≪ρ0σ and ρ0σ≪ℙΩ.P _ _0^σ and W_ _0^σ . By the classical Girsanov theorem [16, Theorem 8.6.6], βtΩ:=βt−∫0tαss _t := _t- _0^t _s\,ds (17) is a standard ℝnR^n-valued Brownian motion under ℙΩP . Equivalently, we get dβt=dβtΩ+αtdt.d _t=d _t + _t\,dt. (18) From the stochastic anti-development in (9), we have dRt=σUt∘dβt.dR_t=σ U_t _t. (19) Substituting the expression for dβtd _t from (18) into (19) yields dRt _t =σUt∘(dβtΩ+αtdt)=σUt∘dβtΩ+σUtαtdt. =σ U_t (d _t + _t\,dt )=σ U_t _t +σ U_t _t\,dt. (20) Since the second term is of finite variation, no Stratonovich correction arises from the change dβt↦dβtΩ+αtdtd _t _t + _t\,dt. Using the definition of αt _t in (11) and of bΩb in (10), we have σUtαt=bΩ(Rt,t).σ U_t _t=b (R_t,t). Plugging this in (20) gives dRt=bΩ(Rt,t)dt+σUt∘dβtΩ.dR_t=b (R_t,t)\,dt+σ U_t _t . (21) The infinitesimal generator of (21) is ℒtΩ=bΩ(⋅,t)+σ22Δ,L_t =b (·,t)+ σ^22 _ G, where bΩb is understood as the corresponding first-order differential operator. On the other hand, the Stratonovich SDE dRt=bΩ(Rt,t)dt+σ∑i=1nXi(Rt)∘dWtidR_t=b (R_t,t)\,dt+σ _i=1^nX_i(R_t) _t^i has generator bΩ(⋅,t)+σ22∑i=1nXi2 b (·,t)+ σ^22 _i=1^nX_i^2 =bΩ(⋅,t)+σ22Δ. =b (·,t)+ σ^22 _ G. (22) Thus, the horizontal representation (21) and the left-invariant representation (12) have the same infinitesimal generator. By the assumed well-posedness in law of the controlled SDE (12) (Assumption 2), their path laws coincide. Consequently, ℙΩP is precisely the path law of the controlled diffusion (12). Since ZtZ_t is a martingale with respect to the filtration that contains R0R_0, we also have ρ0σ[Z1∣R0]=1.E_W_ _0^σ[Z_1 R_0]=1. Hence the change of measure leaves the initial distribution unchanged: R0∼ρ0dμunder ℙΩ.R_0 _0\,dμ\;\;under P . Since bΩ(Rt,t)=RtΩ(Rt,t),b (R_t,t)= E_R_t (R_t,t), we obtain Ut−1bΩ(Rt,t) U_t^-1b (R_t,t) =Ut−1RtΩ(Rt,t)=tΩ(Rt,t). =U_t^-1 E_R_t (R_t,t)= O_t (R_t,t). Since t O_t is orthogonal, ‖αt‖22=1σ2‖Ω(Rt,t)‖22.\| _t\|_2^2= 1σ^2\| (R_t,t)\|_2^2. (23) Substituting αt=1σtΩ(Rt,t) _t= 1σ O_t (R_t,t) into the definition of Z1Z_1 and using (23) give (16), which completes the proof. ∎ I-E Relative Entropy and Control Energy Lemma 1 helps to establish a link between relative entropy and control energy, formalized in Corollary 1. This will be useful in the development that follows. Corollary 1 (Relative entropy and control energy). Under the hypotheses of Lemma 1, suppose additionally that ℙΩ[∫01‖Ω(Rt,t)‖22t]<∞.E_P [ _0^1\| (R_t,t)\|_2^2\,dt ]<∞. (24) Then the KL divergence DKL(ℙΩ∥ρ0σ)D_KL (P _ _0^σ ) is given by DKL(ℙΩ∥ρ0σ)=12σ2ℙΩ[∫01∥Ω(Rt,t)∥22dt].D_KL (P _ _0^σ )= 12σ^2E_P [ _0^1\| (R_t,t)\|_2^2\,dt ]. (25) Proof. From (17), we have dβt=dβtΩ+αtdt.d _t=d _t + _t\,dt. Consequently, (16) gives logdℙΩdρ0σ dP dW_ _0^σ =∫01⟨αt,dβt⟩−12∫01‖αt‖22t = _0^1 _t,d _t - 12 _0^1\| _t\|_2^2\,dt =∫01⟨αt,dβtΩ⟩+12∫01‖αt‖22t. = _0^1 _t,d _t + 12 _0^1\| _t\|_2^2\,dt. (26) Under the additional finite energy assumption (24), relation (23) implies ℙΩ[∫01‖αt‖22t]<∞.E_P [ _0^1\| _t\|_2^2\,dt ]<∞. Therefore, ∫0t⟨αs,dβsΩ⟩ _0^t _s,d _s is a square-integrable martingale under ℙΩP , and hence ℙΩ[∫01⟨αt,dβtΩ⟩]=0.E_P [ _0^1 _t,d _t ]=0. Taking ℙΩP -expectations in (26) gives DKL(ℙΩ∥ρ0σ) D_KL (P _ _0^σ ) =ℙΩ[logdℙΩdρ0σ] =E_P [ dP dW_ _0^σ ] =12ℙΩ[∫01‖αt‖22t] = 12E_P [ _0^1\| _t\|_2^2\,dt ] =12σ2ℙΩ[∫01‖Ω(Rt,t)‖22t], = 12σ^2E_P [ _0^1\| (R_t,t)\|_2^2\,dt ], (27) which proves (25). ∎ I-F Equivalence of the Two Formulations For the equivalence below, we consider admissible controls satisfying Assumptions 1–3. With =C([0,1],),X=C([0,1]; G), equipped with its Borel σ-algebra, let ℳ()M(X) denote the collection of all probability measures on X. Let ρ0σ∈ℳ()W_ _0^σ (X) denote the Wiener measure on X induced by the uncontrolled diffusion dRt=σ∑i=1n(LRt)eei∘dWti,R0∼ρ0dμ.dR_t=σ _i=1^n( DL_R_t)_ee_i g _t^i, R_0 _0\,dμ. (28) For any admissible pair (ρ,Ω)∈01()× (ρ, ) _01( G)×V in Problem 1, let ℙΩ∈ℳ()P (X) denote the path law on X of the corresponding controlled diffusion (2). Since the time-t marginal of ℙΩP has density ρ(⋅,t)ρ(·,t) with respect to μ, we have ℙΩ[∫01‖Ω(Rt,t)‖22t] _P [ _0^1\| (R_t,t)\|_2^2\,dt ] =∫01∫‖Ω(R,t)‖22ρ(R,t)μ(R)t. = _0^1 _ G\| (R,t)\|_2^2ρ(R,t)\,dμ(R)\,dt. (29) Since every admissible control Ω∈ has finite control energy and is assumed to satisfy Assumptions 1– 3, Corollary 1 applies and yields DKL(ℙΩ∥ρ0σ)=12σ2∫01∫∥Ω(R,t)∥22ρ(R,t)dμ(R)dt.D_KL (P _ _0^σ )= 12σ^2 _0^1 _ G\| (R,t)\|_2^2ρ(R,t)\,dμ(R)\,dt. (30) Next, we define Π01 _01 :=∈ℳ()| has marginals ρ0dμ := \M (X)\, |\,M has marginals _0\,dμ and ρ1dμ at t=0,1. and _1\,dμ at t=0,1 \. (31) Since every admissible measure in Π01 _01 possesses the prescribed endpoint marginals, minimizing (7a) is equivalent to minimizing the relative entropy infℙ∈Π01DKL(ℙ∥ρ0σ). _P∈ _01D_KL (P _ _0^σ ). (32) I Existence-Uniqueness of the SBP Solution As an analytic implication of the equivalence established above, we now state and prove the existence-uniqueness result for the SBP. Theorem 1 (Existence-uniqueness of the SBP solution on G). Let G be a compact connected Lie group with Lie algebra g of dimension n, endowed with the left-invariant Riemannian metric induced by ⟨⋅,⋅⟩ ·,· _ g, and let σ>0σ>0. Suppose that the prescribed endpoint densities ρ0,ρ1 _0, _1 satisfy ρ0,ρ1∈2()∩C2(),ρ0(R)>0,ρ1(R)>0,∀R∈. _0, _1 _2( G)\,∩\,C^2( G),\, _0(R)>0,\, _1(R)>0,\,∀\,R∈ G. Assume further that every admissible control Ω∈ satisfies Assumptions 1– 3. Then Problem 1 admits a unique minimizing pair (ρσopt,Ωσopt). ( _σ^opt, _σ^opt ). Proof. We first show that this relative entropy minimization problem has finite value. Let ptσ(R,S)p_t^σ(R,S) denote the transition density, with respect to μ, of the uncontrolled diffusion (28). Since G is compact and connected, ptσp_t^σ is smooth and strictly positive for every t>0t>0. In particular, for t=1t=1, there exist constants 0<cσ≤Cσ<∞0<c_σ≤ C_σ<∞ such that cσ≤p1σ(R,S)≤Cσ,∀R,S∈.c_σ≤ p_1^σ(R,S)≤ C_σ, ∀\,R,S∈ G. (33) The joint law of (R0,R1)(R_0,R_1) under ρ0σW_ _0^σ is therefore 01σ(dR,dS)=ρ0(R)p1σ(R,S)dμ(R)dμ(S).W_01^σ(dR,dS)= _0(R)\,p_1^σ(R,S)\,dμ(R)\,dμ(S). (34) Consider now the product coupling of the prescribed endpoint distributions, π(dR,dS)=ρ0(R)ρ1(S)dμ(R)dμ(S).π(dR,dS)= _0(R) _1(S)\,dμ(R)\,dμ(S). (35) Since p1σ(R,S)>0p_1^σ(R,S)>0, we have π≪01σπ _01^σ, with dπd01σ(R,S)=ρ1(S)p1σ(R,S). dπdW_01^σ(R,S)= _1(S)p_1^σ(R,S). (36) Since ρ1 _1 is continuous and strictly positive on the compact manifold G, there exist constants 0<m1≤M1<∞0<m_1≤ M_1<∞ such that m1≤ρ1(S)≤M1,for allS∈.m_1≤ _1(S)≤ M_1,\;for all\;\,S∈ G. Combining this with (33), the function ρ1(S)/p1σ(R,S) _1(S)/p_1^σ(R,S) is bounded above and away from zero on × G× G. Consequently, DKL(π∥01σ)<∞.D_KL (π _01^σ )<∞. Define a probability measure ℙ¯ P on X by dℙ¯dρ0σ=ρ1(R1)p1σ(R0,R1). d PdW_ _0^σ= _1(R_1)p_1^σ(R_0,R_1). (37) Indeed, using the joint endpoint law (34), ρ0σ[ρ1(R1)p1σ(R0,R1)] _W_ _0^σ [ _1(R_1)p_1^σ(R_0,R_1) ] =∫×ρ1(S)p1σ(R,S)ρ0(R)p1σ(R,S)μ(R)μ(S)=1, = _ G× G _1(S)p_1^σ(R,S) _0(R)p_1^σ(R,S)\,dμ(R)\,dμ(S)=1, so (37) defines a probability measure. Moreover, the joint law of (R0,R1)(R_0,R_1) under ℙ¯ P is precisely π. Therefore ℙ¯∈Π01. P∈ _01. Furthermore, DKL(ℙ¯∥ρ0σ) D_KL ( P _ _0^σ ) =ℙ¯[logρ1(R1)p1σ(R0,R1)] =E_ P [ _1(R_1)p_1^σ(R_0,R_1) ] =DKL(π∥01σ)<∞. =D_KL (π _01^σ )<∞. (38) Consequently, infℙ∈Π01DKL(ℙ∥ρ0σ)<∞. _P∈ _01D_KL (P _ _0^σ )<∞. The relative entropy ℙ↦DKL(ℙ∥ρ0σ)P D_KL (P _ _0^σ ) is lower semicontinuous with respect to weak convergence of probability measures, and its sublevel sets are weakly compact. Moreover, Π01 _01 is weakly closed. Consequently, the path space problem (32) admits a minimizer ℙopt∈Π01P^opt∈ _01. Furthermore, Π01 _01 is convex and DKL(⋅∥ρ0σ)D_KL (· _ _0^σ ) is strictly convex on ℳ()M(X). Hence this minimizer is unique. Since every admissible control induces a measure in Π01 _01, (32) is initially a relaxation of the original control problem. By the standard Schrödinger bridge characterization [17], however, its minimizer is a Markov diffusion obtained through a Doob h-transform, whose drift is generated by a positive Schrödinger potential. Under the present assumptions and compactness of G, this drift is bounded and defines an admissible finite-energy control. So the relaxation is exact, and the minimizer induces a minimizing pair for Problem 1. ∎ Remark 1. It is instructive to juxtapose the existence-uniqueness proof above with that in our prior work [10, Theorem 1]. Beyond techniques, the nature of the arguments are different: [10, Theorem 1] only established the existence-uniqueness of solution a posteriori since then, we had to proceed formally with necessary conditions for optimality, transform those conditions into new variables (Schrödinger system), and then established existence-uniqueness for the resulting system. In contrast, the existence-uniqueness proof for the path space formulation established here is direct. I-A Large-Deviation Interpretation The path space formulation also gives a large-deviation11 1 The DKLD_KL serves as the rate function. interpretation of the Schrödinger bridge problem. Indeed, consider N independent realizations R(1),…,R(N)R^(1),…,R^(N) of the uncontrolled diffusion with common law ρ0σW_ _0^σ, and define the empirical path measure LN:=1N∑k=1NδR(k).L_N:= 1N _k=1^N _R^(k). By Sanov’s large-deviation principle, the probability that LNL_N is close to a path measure ℙP behaves, at the exponential scale exp(−NDKL(ℙ∥ρ0σ)). \! (-ND_KL (P _ _0^σ ) ). Therefore, among all path measures satisfying the prescribed endpoint constraints, the minimizer ℙoptP^opt is the least unlikely, or equivalently the most probable, deviation from the uncontrolled dynamics. Since ℙoptP^opt corresponds to (ρσopt,Ωσopt) ( _σ^opt, _σ^opt ), the optimal Schrödinger bridge may thus be interpreted as the most likely stochastic evolution connecting the two prescribed marginals. IV Static Sinkhorn and Numerical Example Fig. 1: Evolution of the Schrödinger bridge probability density between endpoint distributions ρ0 _0 and ρ1 _1 on the torus 2T^2. In blue is shown the continuous density function ρσopt(⋅,t)ρ^opt_σ(·,t) – the solution of (7) computed by solving the boundary-coupled system of heat PDEs on the Lie group as in [10]. In orange are the scattered data points transported along ℙoptP^opt – the minimizer of (32) computed by solving the static Sinkhorn recursions. To practically demonstrate the equivalence between the solutions of (7) and (32), we solve both formulations for the following antipodal unimodal reference measures ρ0 _0, ρ1 _1 supported on the torus 2T^2: ρ0(θ1,θ2) _0( _1, _2) =exp(25cos(θ1−π3)+30cos(θ2−π2))4π2I0(25)I0(30), = (25 ( _1- π3)+30 ( _2- π2) )4π^2I_0(25)\,I_0(30), ρ1(θ1,θ2) _1( _1, _2) =exp(25cos(θ1−11π8)+30cos(θ2−π2))4π2I0(25)I0(30), = (25 ( _1- 11π8)+30 ( _2- π2) )4π^2I_0(25)\,I_0(30), where the Bessel function I0(κ):=∑m=0∞1(m!)2(κ2)2mI_0(κ):= _m=0^∞ 1(m!)^2 ( κ2 )^2m. To solve (7), the domain 2T^2 for both measures is discretized into a 128×128128× 128 grid, whereover a Schrödinger system (boundary coupled system of forward and backward heat PDEs over the Lie group) is solved by dynamic Sinkhorn recursions; see [10]. For solving (32), we make use of strict convexity of the objective thereof, proven in Theorem 1. The resulting strong duality allows for the solution of (32) via static Sinkhorn recursions over the Lagrange multipliers associated with the constraints of Π01 _01. For the static Sinkhorn recursions, the guarantee of linear convergence to a unique set of multipliers is well-established; see e.g., [18, Sec. V]. Following convergence, the minimizer ℙoptP opt can be obtained from the multipliers. In our implementation, we sample N=50N=50 points from the discretized ρ0 _0 and ρ1 _1 and solve (32) therebetween22 2 The choice to solve (32) over scattered data is for numerical convenience of static Sinkhorn; one can use the same fixed grid as for the solution of (7).. In Figure 1 we see the SB between the two measures is the same when solving both formulations. The 50 sampled data points used in the solution of (32) are shown to track exactly the continuous density function, which is the solution of (7). More simulation and animations are available at: https://gradslab.github.io/LargeDeviationSBP/ V Concluding Remarks We established equivalence between the stochastic optimal control and path space formulations of the SBP on a compact connected Lie group. Using horizontal lift and stochastic anti-development, we derived a Girsanov type change of measure, and related the expected control energy to path space relative entropy. 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