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The Risk-Sensitive Schrödinger Bridge: Is Not a KL Projection
Hamidreza Behjoo
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
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Summary
This paper demonstrates that the classical Schrödinger bridge problem, which relies on a Kullback-Leibler (KL) projection structure solvable by alternating projections, fails when extended to risk-sensitive control using an entropic risk measure. The authors prove that the resulting fixed-point bridge value cannot be represented as a constrained KL minimum against any fixed path-space reference with a regular endpoint law. The obstruction is explicitly computed as a violation of a heat equation by exactly theta/2 for Gaussian bridges. The theory is reformulated using a terminal-multiplier fixed point and an asymmetric factorization of the optimal density.
Entities (8)
Relation Signals (6)
Risk-Sensitive Schrödinger Bridge → lacks → KL Projection
confidence 95% · This letter shows that the fact does not survive risk sensitivity.
Schrödinger bridge → relieson → KL Projection
confidence 95% · The Schrödinger bridge owes its computational power to a single structural fact: by Girsanov's theorem the controlled problem is a Kullback--Leibler (KL) projection
Entropic Risk Measure → replaces → Expected Path Cost
confidence 92% · When the expected path cost is replaced by the entropic risk measure
Gaussian Bridge → violates → Heat Equation
confidence 90% · the Gaussian bridge value violates, by exactly θ/2, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey.
Risk-Sensitive Schrödinger Bridge → uses → Terminal-Multiplier Fixed Point
confidence 85% · In place of the projection, the theory rests on a terminal-multiplier fixed point
Risk-Sensitive Schrödinger Bridge → uses → Asymmetric Factorisation
confidence 85% · and an asymmetric factorisation penalising the score energy of the backward factor.
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Abstract
Abstract:The Schrödinger bridge owes its computational power to a single structural fact: by Girsanov's theorem the controlled problem is a Kullback--Leibler (KL) projection onto a fixed reference measure, solvable by alternating projections. This letter shows that the fact does not survive risk sensitivity. When the expected path cost is replaced by the entropic risk measure and both endpoint marginals are kept as hard constraints, the resulting fixed-point bridge value $J_\theta$ (the soft-problem value at the multiplier that enforces the terminal constraint) admits no representation as a constrained KL minimum against any fixed path-space reference with a regular endpoint law (a class strictly larger than the uniformly elliptic diffusion references: no Markov property is required), even allowing an additive normalisation depending on the initial marginal. Moreover, no single reference generates the one-parameter family in the risk parameter. The obstruction is computed in closed form: the Gaussian bridge value violates, by exactly $\theta/2$, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey. In place of the projection, the theory rests on a terminal-multiplier fixed point and an asymmetric factorisation penalising the score energy of the backward factor.
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- Source: https://arxiv.org/abs/2609.27250v1
- Canonical: https://arxiv.org/abs/2609.27250v1
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The Risk-Sensitive Schrödinger Bridge Is Not a KL Projection Hamidreza Behjoo †thanks: H.˜Behjoo is with the Institute of Systems Medicine, Chinese Academy of Medical Sciences, Suzhou 215123, China (e-mail: hamidreza.behjoo@gmail.com). Abstract The Schrödinger bridge owes its computational power to a single structural fact: by Girsanov’s theorem the controlled problem is a Kullback–Leibler (KL) projection onto a fixed reference measure, solvable by alternating projections. This letter shows that the fact does not survive risk sensitivity. When the expected path cost is replaced by the entropic risk measure and both endpoint marginals are kept as hard constraints, the resulting fixed-point bridge value JθJ_θ (the soft-problem value at the multiplier that enforces the terminal constraint) admits no representation as a constrained KL minimum against any fixed path-space reference with a regular endpoint law (a class strictly larger than the uniformly elliptic diffusion references: no Markov property is required), even allowing an additive normalisation depending on the initial marginal. Moreover, no single reference generates the one-parameter family in the risk parameter. The obstruction is computed in closed form: the Gaussian bridge value violates, by exactly θ/2θ/2, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey. In place of the projection, the theory rests on a terminal-multiplier fixed point and an asymmetric factorisation penalising the score energy of the backward factor. Index Terms: Schrödinger bridge, risk-sensitive control, entropic risk measure, KL duality. I Introduction The Schrödinger bridge problem lies at the intersection of stochastic control, optimal transport, and generative modelling [1, 2]. Its computational power rests on a single structural fact: by Girsanov’s theorem the controlled problem is a Kullback–Leibler (KL) projection onto a fixed reference measure, and can therefore be solved by alternating projections or Sinkhorn iteration in the static formulation. This letter shows that the projection breaks when the expected path cost is replaced by the entropic risk measure 1θlogeθcost 1θ \,e^θ\,cost, θ∈(0,1)θ∈(0,1), with both endpoint marginals kept as hard constraints. The proof computes the obstruction explicitly: the Gaussian bridge value violates, by exactly θ/2θ/2, a heat equation that the endpoint law of any regular reference must obey (Section IV). Risk sensitivity alone would destroy nothing: for a fixed path cost the entropic objective differs from a linear functional of the path measure only by a monotone transformation, so the constrained problem remains a linear program with the usual Schrödinger multiplier structure. The obstruction here is that the kinetic cost is a functional of the controlled law itself rather than a fixed path functional; jointly with the hard endpoint constraints, this destroys the KL-projection structure on which the classical and computational theory rests (Section IV). Two values must be kept distinct throughout: the fixed-point bridge value JθJ_θ of (7), developed here, and the hard-constrained risk-sensitive value θJ_θ of (8), a different object for θ>0θ>0. The two coincide at θ=0θ=0 and separate at first order by an explicit variance–covariance gap (9). Contributions (i) a formulation of the risk-sensitive bridge in which the hard terminal constraint is enforced by a terminal-multiplier fixed point, with a gauge-invariant bridge value JθJ_θ (Section I); (i) a three-step impossibility proof (the per-θ KL representation is circular, no reference is uniform in θ, and no fixed endpoint-regular reference represents JθJ_θ at all), with the obstruction computed explicitly as a θ/2θ/2 heat-equation defect (Section IV); (i) a closed-form Riccati reduction for Gaussian targets to decoupled scalar shooting equations, proving well-posedness of the fixed point in that setting (Section V); and (iv) the structure that replaces the projection: an asymmetric factorisation p∗=φψ1/(1−θ)p^*= \,ψ^1/(1-θ) penalising the score energy of the backward factor (Section VI). Related work Exponential-of-integral cost criteria go back to [3] and were developed in generality by [4, 5]; the equivalence with a minimax game against an entropy-penalised adversary is central to robust control [6], and risk-sensitive extensions within linearly solvable and path-integral control appear in [7]. All of these consider unconstrained problems: the terminal law is free, and the two hard marginal constraints responsible for our obstruction are absent. Pricing the log-likelihood rather than the kinetic energy [8] yields a Rényi divergence: a convex program, but again with a free terminal law. Existing “robust” bridge variants relax the hard terminal constraint [9] or address parameter perturbations; none modifies the path-cost objective, and all retain the KL-projection structure. To the best of our knowledge this is the first formulation of the Schrödinger bridge under the entropic risk measure with hard endpoint constraints, and the first characterisation of the resulting breakdown of KL/I-projection duality. I Preliminaries: the standard bridge Throughout, α:=1−θα:=1-θ, (m,Σ)N(m, ) denotes the Gaussian law, and ⟨f,μ⟩:=∫fμ f,μ := f\,dμ. Fix the horizon [0,1][0,1] and consider the controlled diffusion in ℝdR^d dXt=(ft(Xt)+ut(Xt))dt+κtdWt,X0∼μ0,dX_t= (f_t(X_t)+u_t(X_t) )\,dt+ _t\,dW_t, X_0 _0, (1) with given drift ftf_t, scalar κt>0 _t>0, Markov control u, and PuP^u the path law on C([0,1],ℝd)C([0,1];R^d); P0P^0 is the uncontrolled reference. The standard Schrödinger bridge [1] is infuPu[∫01(12κt‖ut(Xt)‖2+Vt(Xt))t]s.t.X1∼μ1. _u\;E^P^u\! [ _0^1\! ( 12 _t\|u_t(X_t)\|^2+V_t(X_t) )dt ] .t.\;X_1 _1. (2) By Girsanov’s theorem the objective equals KL(Pu∥PV)KL(P^u\|P^V) up to an additive constant, where dPV/dP0∝exp(−∫01Vt(Xt)dt)dP^V/dP^0 (- _0^1V_t(X_t)\,dt); the bridge is therefore the static problem infQ:Q0=μ0,Q1=μ1KL(Q∥PV) _Q:\,Q_0= _0,\,Q_1= _1KL(Q\|P^V), an I-projection onto a fixed reference. The value function S0S_0 of the soft problem associated with (2), the θ=0θ=0 member of the risk-sensitive family introduced in Section I, satisfies −∂tS0=Vt+ft⋅∇S0+κt2ΔS0−κt2‖∇S0‖2,S0(1,x)=λ(x), split- _tS_0&=V_t+f_t·∇ S_0+ _t2 S_0- _t2\|∇ S_0\|^2,\\ S_0(1,x)&=λ(x), split (3) with optimal feedback u∗=−κt∇S0u^*=- _t∇ S_0, and the Cole–Hopf transform S0=−logψS_0=- ψ linearises (3) to −∂tψ+Vψ=ft⋅∇ψ+κt2Δψ,ψ(1,x)=e−λ(x).- _tψ+Vψ=f_t·∇ψ+ _t2 ψ, ψ(1,x)=e^-λ(x). (4) With the forward factor φ solving ∂tφ=−∇⋅(ftφ)+κt2Δφ−Vφ,φ(0,⋅)=μ0ψ(0,⋅), _t =-∇·(f_t )+ _t2 -V , (0,·)= _0ψ(0,·), (5) the optimally controlled density admits the symmetric factorisation p∗=φψp^*= ψ. We work formally with classical positive solutions of the forward–backward system under standard parabolic regularity; the Gaussian case of Section V is fully self-contained. I The risk-sensitive bridge Replace the linear expectation in (2) by the cumulant-generating function. For a prescribed terminal cost λ, the soft risk-sensitive value is Sθ(λ):=infu1θlogPu[exp(θ∫01(‖ut‖22κt+Vt(Xt))t+θλ(X1))], splitS_θ(λ):= _u 1θ ^P^u\! [ (&θ\! _0^1\! ( \|u_t\|^22 _t+V_t(X_t) )dt\\ &+θλ(X_1) ) ], split (6) with only the initial constraint imposed; at θ=0θ=0 this is the classical soft value, and with the terminal constraint restored the bridge value (7) recovers (2) exactly. The bridge problem imposes in addition the hard terminal constraint X1∼μ1X_1 _1, and λ is then no longer a datum: it becomes a terminal multiplier, i.e. a shooting parameter, enforcing the constraint. I-A Gauge and the bridge value The soft value is exactly shift-equivariant, Sθ(λ+c)=Sθ(λ)+cS_θ(λ+c)=S_θ(λ)+c for c∈ℝc , while the terminal coupling (18) below is blind to constants; the multiplier is determined only up to the coset λ∗+ℝλ^*+R, and the optimally controlled process is untouched by the constant. The gauge-invariant combination is the difference Jθ(μ0→μ1):=Sθ(λ∗)−⟨λ∗,μ1⟩,J_θ( _0→ _1):=S_θ(λ^*)- λ^*, _1 , (7) with λ∗λ^* any representative of the coset. Definition 1 (Bridge value). The risk-sensitive bridge value is the gauge-invariant fixed-point value (7). For computations we fix the representative by the gauge ⟨λ∗,μ1⟩=0 λ^*, _1 =0, under which Jθ=Sθ(λ∗)J_θ=S_θ(λ^*); the definition does not depend on the gauge. The definition is canonical in two senses. At θ=0θ=0 the terminal coupling becomes the classical Schrödinger system and (7) is precisely the Legendre identity J0=supλS0(λ)−⟨λ,μ1⟩J_0= _λ\S_0(λ)- λ, _1 \: the classical bridge value is recovered with no arbitrary choices. Second, the fixed point is well-posed at least in the Gaussian setting, by strict monotonicity of the scalar shooting map (Remark 2). For θ>0θ>0 the Legendre identity fails: the first variation of λ↦Sθ(λ)−⟨λ,μ1⟩λ S_θ(λ)- λ, _1 at the fixed point is q1θ−μ1q_1^θ- _1, with q1θq_1^θ a tilted terminal law, so the fixed point is not a saddle point of a Lagrangian. I-B The hard value and the first-order gap For θ>0θ>0, (7) is not the value of the hard-constrained risk-sensitive control problem θ(μ0→μ1):=infu:X0∼μ0,X1∼μ11θlogu[eθKu],Ku=∫01(‖ut‖22κt+Vt)dt. splitJ_θ( _0→ _1):= _u:\,X_0 _0,\,X_1 _1& 1θ ^u\! [e^θ K^u ],\\ K^u&= _0^1 ( \|u_t\|^22 _t+V_t )dt. split (8) The two coincide at θ=0θ=0 and separate at first order: writing u0∗u_0^* for the classical bridge control and λ0∗ _0^* for the classical multiplier, a cumulant expansion with an envelope argument gives Jθ−θ=θ(12Varu0∗(λ0∗(X1))+Covu0∗(K,λ0∗(X1)))+o(θ),J_θ-J_θ=θ ( 12Var^u_0^*\! ( _0^*(X_1) )+Cov^u_0^*\! (K,\, _0^*(X_1) ) )+o(θ), (9) generically nonzero, of either sign. The same expansion applied to JθJ_θ alone gives Jθ=J0+θ2Varu0∗(K+λ0∗(X1))+o(θ),J_θ=J_0+ θ2\,Var^u_0^*\! (K+ _0^*(X_1) )+o(θ), (10) and (9) is its difference with the companion expansion of θJ_θ. The two problems differ because the kinetic cost is a functional of the controlled law, not a fixed path functional: in (8) the constraint is on the law of X1X_1, so its KKT multiplier is not absorbed into the path cost and acts outside the entropic exponential, and (8) is not a linear program over path measures. The fixed-point construction (7) instead keeps the terminal potential inside the exponential, which is precisely what makes the generalised Cole–Hopf transform below available. I-C HJB equation and generalised Cole–Hopf By dynamic programming, the state-conditioned soft value Sθ(t,x)S_θ(t,x) satisfies −∂tSθ=Vt+ft⋅∇Sθ+κt2ΔSθ−κt2(1−θ)‖∇Sθ‖2,Sθ(1,x)=λ(x). split- _tS_θ&=V_t+f_t·∇ S_θ+ _t2 S_θ- _t2(1-θ)\|∇ S_θ\|^2,\\ S_θ(1,x)&=λ(x). split (11) The generalised Cole–Hopf transform ψt(x):=exp(−(1−θ)Sθ(t,x))⟺Sθ(t,x)=−logψt(x)1−θ _t(x):= \! (-(1-θ)S_θ(t,x) )\; \;S_θ(t,x)=- _t(x)1-θ (12) linearises (11) to the backward PDE −∂tψt+(1−θ)Vtψt=ft⋅∇ψt+κt2Δψt,ψ(1,x)=e−(1−θ)λ(x), split- _t _t+(1-θ)V_t _t&=f_t·∇ _t+ _t2 _t,\\ ψ(1,x)&=e^-(1-θ)λ(x), split (13) whose generator is θ-independent. The optimal feedback is ut∗(x)=κt1−θ∇logψt(x).u_t^*(x)= _t1-θ∇ _t(x). (14) I-D Asymmetric factorisation Under (14), the controlled density p∗p^* satisfies the Fokker–Planck equation with drift ft+κt1−θ∇logψtf_t+ _t1-θ∇ _t. Proposition 1 (Forward decoupling). Let ψ solve (13) and let p∗p^* be the optimally controlled density. Then p∗(t,x)=φ(t,x)ψ(t,x)11−θ,p^*(t,x)= (t,x)\,ψ(t,x) 11-θ, (15) where φ satisfies the closed linear forward equation ∂tφ=−∇⋅(ftφ)+κt2Δφ−Veff(t,x)φ, _t =-∇·(f_t )+ _t2 -V_ eff(t,x) , (16) with effective potential Veff(t,x)=V(t,x)+κtθ2(1−θ)2‖∇logψ(t,x)‖2,V_ eff(t,x)=V(t,x)+ _tθ2(1-θ)^2\|∇ ψ(t,x)\|^2, (17) and initial condition φ(0,x)=μ0(x)/ψ(0,x)1/(1−θ) (0,x)= _0(x)/ψ(0,x)^1/(1-θ). Proof. Insert the ansatz p∗=φψrp^*= ψ^r into the Fokker–Planck equation. The drift divergence produces a cross term proportional to ∇φ⋅∇ψ∇ ·∇ψ with coefficient r−(1−θ)−1r-(1-θ)^-1; setting it to zero yields r=1/(1−θ)r=1/(1-θ). Dividing by ψrψ^r and using (13) to eliminate ∂tψ _tψ and Δψ ψ, the remaining scalar terms combine to give (16)–(17). ∎ Since θ≥0θ≥ 0, (17) gives Veff≥V_ eff≥ V pointwise, strictly wherever ∇logψ≠0∇ ψ≠ 0: risk sensitivity can only add to the potential landscape seen by the forward factor, concentrated where the backward score is steep. The hard terminal constraint p∗(1,⋅)=μ1p^*(1,·)= _1 reads φ(1,x)ψ(1,x)1/(1−θ)=μ1(x), (1,x)\,ψ(1,x)^1/(1-θ)= _1(x), (18) equivalently φ(1,⋅)e−λ∗(⋅)=μ1 (1,·)\,e^-λ^*(·)= _1 in terms of the multiplier: the compact identity that makes λ∗λ^* a multiplier for the hard constraint, determined implicitly as a shooting parameter. At θ=0θ=0 the score-energy term vanishes and (16)–(18) reduce to (5) and the symmetric factorisation. IV The fixed-point value has no KL dual For the standard bridge, Girsanov yields Pu∫01(‖ut‖22κt+Vt)dt=KL(Pu∥PV)+constE^P^u\! _0^1( \|u_t\|^22 _t+V_t)\,dt=KL(P^u\|P^V)+const, so the bridge is the I-projection infQ:Q0=μ0,Q1=μ1KL(Q∥PV) _Q:\,Q_0= _0,\,Q_1= _1KL(Q\|P^V). It is natural to ask whether JθJ_θ, for θ>0θ>0, admits an analogous projection representation. It does not, in a precise sense. The statements concern the fixed-point value JθJ_θ; the hard value θJ_θ likewise admits no linear-programming formulation, for the independent reason that its exponent is law-dependent. The structural obstruction is the asymmetric factorisation (15): the Gibbs form dQ∗/dR∝φ⋅ψdQ^*/dR ·ψ, with factors of equal weight, is exactly what underlies the static KL dual, and it is what θ>0θ>0 breaks. The asymmetry alone is not a proof: Lemma 1 shows that a solution-dependent reference reproduces the risk-sensitive bridge optimizer in full; the impossibility is argued at the level of the value. Theorem 1 (No KL dual). Fix the problem data (f,κ,V)(f,κ,V), let θ∈(0,1)θ∈(0,1), and assume the terminal coupling (18) admits a solution λ∗+ℝλ^*+R for the endpoint pairs under consideration, so that JθJ_θ is defined; this holds in the Gaussian setting, which is what the proofs test against. Then: (i) (circularity) for each fixed θ, a Markov reference whose I-projection value reproduces JθJ_θ exists only if the reference is built from the unknown solution ∇Sθ∇ S_θ itself, via the matching condition (19); (i) (no θ-uniform reference) no single Markov reference represents JθJ_θ as a constrained KL minimum for two distinct values of θ; (i) (no fixed reference at all) there is no endpoint-regular (two-sided Gaussian endpoint joint law, Definition 2) path-space reference RθR_θ, Markovian or not, and no additive correction C(μ0)C( _0), such that Jθ(μ0→μ1)=infQ:Q0=μ0,Q1=μ1KL(Q∥Rθ)+C(μ0)J_θ( _0→ _1)= _Q:\,Q_0= _0,\,Q_1= _1KL(Q\|R_θ)+C( _0) for all absolutely continuous marginals with full support and finite second moment. Part (i) is Lemma 1, part (i) is Proposition 2, and part (i) is Proposition 3; the last two are proved for the scalar Brownian data f≡0f≡ 0, κ=1κ=1, V≡0V≡ 0 of Lemma 2: the obstruction appears already in the simplest setting, which precludes any representation serving general data. IV-A The per-θ representation is circular Consider a candidate Markov reference Ph,UP^h,U with drift h and Feynman–Kac potential U (a sub-probability path measure when U≠0U≠ 0, to which KLKL extends verbatim); relative to it, infQKL(Q∥Ph,U) _QKL(Q\|P^h,U) is itself an ordinary bridge problem whose value function S0h,US_0^h,U satisfies (3) with (f,V)(f,V) replaced by (h,U)(h,U). Lemma 1 (Matching condition). Fix the problem data (f,κ,V)(f,κ,V), a terminal cost λ, and a Markov reference (h,U)(h,U) for which the two HJB equations admit classical solutions. Then S0h,U(⋅,λ)≡Sθ(⋅,λ)S_0^h,U(·\,;λ)≡ S_θ(·\,;λ) if and only if U(t,x)=V(t,x)+(f(t,x)−h(t,x))⋅∇Sθ(t,x)+κtθ2‖∇Sθ(t,x)‖2 splitU(t,x)=V(t,x)&+ (f(t,x)-h(t,x) )·∇ S_θ(t,x)\\ &+ _tθ2\|∇ S_θ(t,x)\|^2 split (19) for all (t,x)(t,x). When (19) holds with h=fh=f, i.e. U=VeffU=V_ eff of (17), and λ=λ∗λ=λ^* implements X1∼μ1X_1 _1 via (18), the I-projection of Pf,VeffP^f,V_ eff onto Q0=μ0,Q1=μ1\Q_0= _0,\,Q_1= _1\ coincides with the risk-sensitive bridge optimizer Pu∗P^u^* itself, not merely with its value. Proof. Subtracting the two HJB equations and using S0≡SθS_0≡ S_θ gives (19); conversely, under (19) the function SθS_θ solves the reference HJB with the same terminal datum, and uniqueness gives S0≡SθS_0≡ S_θ. For the last statement, the reference bridge’s optimal total drift is h−κt∇S0h- _t∇ S_0; with h=fh=f and S0=SθS_0=S_θ this is f+κt1−θ∇logψf+ _t1-θ∇ ψ, exactly (14). Same initial law and same feedback give the same path measure. ∎ So a KL-projection representation of JθJ_θ exists for each fixed θ, but only by building the reference out of ∇Sθ∇ S_θ itself, i.e. out of the unknown solution: a construction that presupposes the answer and is not a duality in any useful sense; read positively, the matching condition made constructive is a fixed-point iteration on the multiplier coset. IV-B Two ingredients The remaining proofs rest on an explicit Gaussian value and a static reduction. Lemma 2 (Point-mass Gaussian value). Let d=1d=1, f≡0f≡ 0, V≡0V≡ 0, κ=1κ=1, μ0=δ0 _0= _0, μ1=(m,s2) _1=N(m,s^2), and write α=1−θα=1-θ. Then vθ(m,s2):=Jθ(δ0→(m,s2))=12αlog(1+a1)−a12α(m2+s2)+m2M−αB2M2m2, splitv_θ(m,s^2)&:=J_θ( _0 (m,s^2))\\ &= 12α (1+a_1)- a_12α (m^2+s^2 )\\ & + m^2M- α B2M^2\,m^2, split (20) where a1a_1 is determined by the point-mass shooting map s2=∫01(1+a1τ)−2/αdτ,s^2= _0^1 (1+a_1τ )^-2/α\,dτ, (21) and M:=∫01(1+a1τ)−1/α−1dτM:= _0^1(1+a_1τ)^-1/α-1\,dτ, B:=∫01(1+a1τ)−2τ=1/(1+a1)B:= _0^1(1+a_1τ)^-2\,dτ=1/(1+a_1). The map a1↦s2a_1 s^2 is a strictly decreasing bijection of (−1,∞)(-1,∞) onto (0,∞)(0,∞), with s2=1⇔a1=0s^2=1 a_1=0. At θ=0θ=0, (20) collapses to 12(m2+s2−1−logs2)=KL((m,s2)∥(0,1)) 12(m^2+s^2-1- s^2)=KL(N(m,s^2)\,\|\,N(0,1)), as it must. Proof. Set k=0k=0 in the scalar Riccati system of Section V: A˙=A2 A=A^2, b˙=Ab b=Ab, whose solutions are rational in t; passing σ(0)↓0σ(0) 0 in the shooting equation (31) yields (21) (cf. Remark 2), and direct evaluation of (7) gives (20). Moreover ∂s2/∂a1=−2α∫01τ(1+a1τ)−2/α−1dτ<0∂ s^2/∂ a_1=- 2α _0^1τ(1+a_1τ)^-2/α-1dτ<0 is nonvanishing, so a1a_1 is a C1C^1 function of s2s^2 down to s2=1s^2=1, with derivative −α-α there. ∎ Definition 2 (Endpoint regularity). A probability measure R on C([0,1],ℝd)C([0,1];R^d) is endpoint-regular if its endpoint law R01:=LawR(X0,X1)R_01:=Law_R(X_0,X_1) has a positive continuous density r(x,y)r(x,y) with two-sided Gaussian bounds: there exist c,Λ>0c, >0 and a continuous m0m_0 of at most linear growth such that c−1e−Λ|y−m0(x)|2≤r(x,y)≤ce−Λ−1|y−m0(x)|2c^-1e^- |y-m_0(x)|^2\;≤\;r(x,y)\;≤\;c\,e^- ^-1|y-m_0(x)|^2 (22) for all x,y∈ℝdx,y ^d. This holds for the path law of any uniformly elliptic diffusion with bounded measurable drift and bounded Feynman–Kac potential by Aronson’s two-sided estimates [10], and for Ornstein–Uhlenbeck references by their explicit Gaussian law. Lemma 3 (Static reduction and the point-mass limit). Let R be endpoint-regular and set IR(μ0,μ1):=infQ:Q0=μ0,Q1=μ1KL(Q∥R)I_R( _0, _1):= _Q:\,Q_0= _0,\,Q_1= _1KL(Q\|R). Then: (i) (static reduction) IR(μ0,μ1)=infπ∈Π(μ0,μ1)KL(π∥R01)I_R( _0, _1)= _π∈ ( _0, _1)KL(π\|R_01), a static Schrödinger problem on ℝd×ℝdR^d×R^d; (i) (point-mass limit) if μ0ε=(0,εId) _0 =N(0, I_d) and μ1,μ1∗ _1, _1^* are absolutely continuous with finite second moment and finite entropy, then, with ρ0(⋅):=R(X1∈⋅∣X0=0) _0(·):=R(X_1∈· X_0=0), IR(μ0ε,μ1)−IR(μ0ε,μ1∗)→ε↓0KL(μ1∥ρ0)−KL(μ1∗∥ρ0).I_R( _0 , _1)-I_R( _0 , _1^*)\; [ 0]\;KL( _1\| _0)-KL( _1^*\| _0). (23) Proof. Part (i) is the chain rule for KL divergence disintegrated over the endpoint pair: the bridge term is minimised at 00 by adopting the bridges of R. Part (i) is a two-sided squeeze under (22): the two-sided Gaussian bounds control the contribution of X0≠0\X_0≠ 0\ as ε↓0 0, and positivity and continuity of r near (0,⋅)(0,·) give the matching lower bound. ∎ Some restriction on the additive freedom C is necessary for the question to be meaningful: if C could depend on both marginals, any reference would represent JθJ_θ by setting C(μ0,μ1):=Jθ−infKL(⋅∥Rθ)C( _0, _1):=J_θ- (·\,\|R_θ). Dependence on μ0 _0 only is the natural class: for the standard bridge the additive term is the Feynman–Kac normalisation, independent of the marginals altogether. IV-C No reference works uniformly in θ Proposition 2 (No θ-uniform KL dual). There is no Markov reference (h,U)(h,U), chosen independently of θ, with h Lipschitz in x of at most linear growth and U bounded, and constants CθC_θ independent of the marginals, such that, in the setting of Lemma 2, Jθ(μ0→μ1)=infQ:Q0=μ0,Q1=μ1KL(Q∥Ph,U)+CθJ_θ( _0→ _1)= _Q:\,Q_0= _0,\,Q_1= _1KL(Q\|P^h,U)+C_θ holds for two distinct values θ∈[0,1)θ∈[0,1) and all absolutely continuous marginals μ0,μ1 _0, _1. Proof. Suppose a fixed pair (h,U)(h,U) satisfies the identity for θ1≠θ2 _1≠ _2; write R=Ph,UR=P^h,U. Apply the identity with μ0ε=(0,ε) _0 =N(0, ) and two Gaussian targets, subtract, and let ε↓0 0. On the left the Riccati formulas of Lemma 2 are continuous in the initial variance down to σ(0)=0σ(0)=0; on the right Lemma 3 applies, since the endpoint kernel of Ph,UP^h,U is endpoint-regular (bounded h by Aronson’s estimate [10]; linear h explicitly Gaussian; general linear-growth Lipschitz h by Girsanov comparison against the driftless kernel and the Cauchy–Schwarz inequality on a finite horizon). Hence, with ρ0(⋅):=R(X1∈⋅∣X0=0) _0(·):=R(X_1∈· X_0=0), vθi(m,s2)=KL((m,s2)∥ρ0)+Cθiv_ _i(m,s^2)=KL(N(m,s^2)\,\|\, _0)+C_ _i for i=1,2i=1,2 and all m∈ℝm , s2>0s^2>0. Subtracting, vθ1−vθ2v_ _1-v_ _2 must be the constant Cθ1−Cθ2C_ _1-C_ _2. But at s2=1s^2=1 (so a1=0a_1=0, M=B=1M=B=1), (20) gives vθ(m,1)=m2−α2m2=1+θ2m2v_θ(m,1)=m^2- α2m^2= 1+θ2m^2, hence vθ1(m,1)−vθ2(m,1)=12(θ1−θ2)m2v_ _1(m,1)-v_ _2(m,1)= 12( _1- _2)m^2, which is not constant in m, a contradiction. ∎ Thus no fixed (h,U)(h,U) generates the family Jθ∈[0,1)\J_θ\_θ∈[0,1) the way the ordinary bridge is a projection onto the single reference PVP^V: the risk-sensitive bridge is a one-parameter family of different control problems, not a re-weighting of one reference measure. IV-D No fixed reference works at all Proposition 3 (No endpoint-regular reference). Let d=1d=1, f≡0f≡ 0, κ=1κ=1, V≡0V≡ 0, and let θ∈(0,1)θ∈(0,1). There exists no endpoint-regular probability measure RθR_θ on C([0,1],ℝ)C([0,1];R) and no map μ0↦C(μ0) _0 C( _0) such that Jθ(μ0→μ1)=infQ:Q0=μ0,Q1=μ1KL(Q∥Rθ)+C(μ0)J_θ( _0→ _1)= _Q:\,Q_0= _0,\,Q_1= _1KL(Q\|R_θ)+C( _0) for all absolutely continuous marginals μ0,μ1 _0, _1 with full support and finite second moment. Proof. Suppose (Rθ,C)(R_θ,C) satisfies the identity. Step 1: point-mass test. Apply the identity with μ0ε=(0,ε) _0 =N(0, ) and two Gaussian targets, and subtract: C cancels since it depends only on μ0 _0. Let ε↓0 0. On the left, the Riccati formulas are continuous down to σ(0)=0σ(0)=0; on the right, Lemma 3(i) applies with ρ0(⋅):=Rθ(X1∈⋅∣X0=0) _0(·):=R_θ(X_1∈· X_0=0), giving vθ(m,s2)=KL((m,s2)∥ρ0)+constv_θ(m,s^2)=KL (N(m,s^2)\,\|\, _0 )+const (24) for all m∈ℝm , s2>0s^2>0, with the constant independent of (m,s2)(m,s^2). Step 2: a deconvolution criterion. Since KL((m,s2)∥ρ)=−12log(2πs2)−12−Y∼(m,s2)[logρ(Y)]KL(N(m,s^2)\|ρ)=- 12 (2π s^2)- 12-E_Y (m,s^2)[ ρ(Y)], the identity (24) says that Lθ(m,s2):=−vθ(m,s2)−12log(2πs2)−12L_θ(m,s^2):=-v_θ(m,s^2)- 12 (2π s^2)- 12 is, up to a constant, the Gaussian smoothing at bandwidth s2s^2 of the fixed function logρ0 _0. Gaussian smoothing obeys the heat equation in (m,s2)(m,s^2): ∂s2Lθ=12∂m2Lθfor all m∈ℝ,s2>0, _s^2L_θ= 12\, _m^2L_θ all m ,\ s^2>0, (25) a necessary condition on the value vθv_θ alone. Step 3: explicit failure at s2=1s^2=1. At s2=1s^2=1 one has a1=0a_1=0, M=B=1M=B=1, so vθ(m,1)=1+θ2m2v_θ(m,1)= 1+θ2m^2 and 12∂m2Lθ(m,1)=−12∂m2vθ(m,1)=−1+θ2 12 _m^2L_θ(m,1)=- 12 _m^2v_θ(m,1)=- 1+θ2. On the other hand vθ(0,s2)=12αlog(1+a1)−a1s22αv_θ(0,s^2)= 12α (1+a_1)- a_1s^22α, and differentiating (21) at a1=0a_1=0 gives da1ds2|s2=1=−α da_1ds^2 |_s^2=1=-α; the two terms of ∂s2vθ(0,s2) _s^2v_θ(0,s^2) then cancel at s2=1s^2=1, giving ∂s2Lθ|(0,1)=−∂s2vθ|(0,1)−12=−12 _s^2L_θ|_(0,1)=- _s^2v_θ|_(0,1)- 12=- 12. The heat equation (25) would require −12=−1+θ2- 12=- 1+θ2, which fails for every θ>0θ>0: the heat-equation defect is exactly θ/2θ/2. Hence no density ρ0 _0, and a fortiori no reference RθR_θ, can satisfy (24). At θ=0θ=0 the two sides agree and (24) holds with ρ0=(0,1) _0=N(0,1), the Brownian endpoint law: the classical KL duality, recovered as a sanity check. ∎ Remark 1 (An inf–sup in place of the KL dual). The structural reason is visible in the Donsker–Varadhan formula [11, 12]: the entropic risk is itself a Legendre object, so JθJ_θ is an inf–sup of KL-type objects (classically, a game against a drift-perturbing adversary [3, 6]), whereas the standard bridge is a single inf; a projection would force the inner supremum to collapse, which happens only at θ=0θ=0. V The Gaussian case: closed-form Riccati reduction Specialise to ft=0f_t=0, κt=κ>0 _t=κ>0, V(x)=k2‖x‖2V(x)= k2\|x\|^2 (k≥0k≥ 0), μ0=(0,Id) _0=N(0,I_d), μ1=(m1,Σ1) _1=N(m_1, _1) with Σ1∈++d _1 _++^d, and define γ:=αk/κγ:= α k/κ, ω:=αkκω:= α kκ. Posit the log-quadratic ansatz ψ(t,x)=exp(−12x⊤A(t)x+b(t)⊤x+c(t))ψ(t,x)= (- 12x A(t)x+b(t) x+c(t)), φ(t,x)=exp(−12x⊤D(t)x+E(t)⊤x+F(t)) (t,x)= (- 12x D(t)x+E(t) x+F(t)) with A,D∈dA,D ^d. The optimally controlled marginal is then Gaussian with precision Σ(t)−1=D(t)+A(t)/α (t)^-1=D(t)+A(t)/α; positivity is required only of this sum. Inserting the ansatz into (13) gives the backward Riccati system A˙=κA2−αkI,b˙=κAb, A=κ A^2-α kI, b=κ Ab, (26) with c˙=κ2(trA−‖b‖2) c= κ2(trA-\|b\|^2). and since ‖∇logψ‖2\|∇ ψ\|^2 is quadratic, the effective potential (17) remains quadratic: the ansatz is self-consistent. The forward equation (16) yields D˙=−κD2+kI+κθα2A2,E˙=−κDE+κθα2Ab, D=-κ D^2+kI+ κθα^2A^2, E=-κ DE+ κθα^2Ab, (27) and in the total precision the kIkI terms cancel: dtΣ(t)−1=−κD2+κα2A2, ddt (t)^-1=-κ D^2+ κα^2A^2, (28) a closed system for (A,Σ−1)(A, ^-1), which recovers the covariance equation Σ˙=κI−κα(AΣ+ΣA), =κ I- κα (A + A ), (29) exactly the moment evolution of the controlled Ornstein–Uhlenbeck SDE with drift −κα(A(t)x−b(t))- κα(A(t)x-b(t)). In the eigenbasis of Σ1 _1, A(t)A(t) and Σ(t) (t) are simultaneously diagonal; each diagonal entry satisfies the scalar system a˙i=κai2−αk,σ˙i=κ−2καaiσi, a_i=κ a_i^2-α k, σ_i=κ- 2καa_i _i, (30) with σi(0)=1 _i(0)=1, σi(1)=ℓi _i(1)= _i. The backward equation has an explicit hyperbolic solution, and the terminal condition reduces to the scalar shooting equation ℓi=γ2/α(gi(1)−2/α+κ∫01gi(τ)−2/αdτ), _i=γ^2/α (g_i(1)^-2/α+κ _0^1g_i(τ)^-2/α\,dτ ), (31) with gi(τ)=γcosh(ωτ)+ai,1sinh(ωτ)g_i(τ)=γ (ωτ)+a_i,1 (ωτ) and ai,1:=ai(1)a_i,1:=a_i(1): one transcendental equation per coordinate, completely decoupled. The mean condition is then a single affine equation per coordinate for bi,1b_i,1. Remark 2 (Well-posedness; Brownian limit). The map ai,1↦ℓia_i,1 _i in (31) is strictly decreasing (for τ>0τ>0, gi(τ)g_i(τ) is strictly increasing in ai,1a_i,1) and surjective onto (0,∞)(0,∞), so a unique root exists for every ℓi>0 _i>0. This proves existence and uniqueness of the terminal-multiplier coset of Definition 1 in the Gaussian setting. In the Brownian case k=0k=0 the hyperbolic formula degenerates to a rational one, and passing σi(0)↓0 _i(0) 0 recovers the point-mass map (21) used in Section IV. The obstruction, in factor language In the same Brownian setting, write the terminal forward factor as φ(1,x)∝exp(−12D(1)x2+E(1)x) (1,x) (- 12D(1)x^2+E(1)x). Precision addition in the coupling (18) and the shooting map (21) give D(1)=1s2−a1α=1+θ3(1−θ)2a12+O(a13)D(1)= 1s^2- a_1α=1+ θ3(1-θ)^2a_1^2+O(a_1^3). For θ>0θ>0 the terminal precision from a point mass varies with the target variance near s2=1s^2=1, whereas a fixed reference forces target-independent endpoint bridges (θ=0θ=0: D(1)≡1D(1)≡ 1). This is the factor-level form of the value-level defect of Proposition 3. Fig. 1: Gaussian bridge with κ=k=1κ=k=1, μ0=(0,1) _0=N(0,1), μ1=(1,1/4) _1=N(1,1/4), computed from (30)–(31). (a) Marginal variance Σ(t) (t): the endpoint constraints are met exactly for every θ, while the intermediate variance contracts as θ grows. (b) Bridge value JθJ_θ: it leaves the classical value J0J_0 with the first-order slope of (10) and diverges at a finite θ, the classical breakdown of the entropic criterion [4]. Numerical illustration Figure 1 shows the one-dimensional case κ=k=1κ=k=1, μ0=(0,1) _0=N(0,1), μ1=(1,1/4) _1=N(1,1/4), solved through the scalar system (30)–(31). The terminal constraint is met exactly at every θ, while the intermediate variance contracts as θ grows. The bridge value leaves the classical value with the first-order slope of (10), then grows superlinearly and diverges at a finite θ: for an unbounded initial law the entropic criterion has a finite breakdown point, beyond which no finite-cost bridge exists. The breakdown attaches to the unbounded initial law rather than to the construction: the state-conditioned soft value Sθ(t,x)S_θ(t,x) and the point-mass case of Lemma 2 remain finite for every θ<1θ<1. VI What replaces the projection The failure of the KL projection is not the failure of structure. For θ>0θ>0 the fixed-point bridge retains the generalised Cole–Hopf linearisation (12)–(13), whose backward generator is θ-independent, and the asymmetric factorisation p∗=φψ1/(1−θ)p^*= \,ψ^1/(1-θ) under the effective potential (17); the value remains an inf–sup of KL-type objects (Remark 1). What is lost is only the single fixed reference and the projection geometry it induces. The mechanism is visible at first order in θ: expanding ψθ=ψ(0)+θψ(1)+O(θ2) _θ=ψ^(0)+θψ^(1)+O(θ^2), φθ=φ(0)+θφ(1)+O(θ2) _θ= ^(0)+θ ^(1)+O(θ^2) around the standard bridge pair and matching the O(θ)O(θ) terms of (13) and (16) gives linear equations with explicit sources: ψ(1)ψ^(1) solves the standard backward PDE with source Vψ(0)Vψ^(0), and φ(1) ^(1) solves the standard forward PDE with the negative source −κt2‖∇logψ(0)‖2φ(0)- _t2\|∇ ψ^(0)\|^2\, ^(0). Proposition 4 (First-order density correction). The optimally controlled density expands as pθ∗(t,x)=φ(0)ψ(0)+θ[φ(0)ψ(0)logψ(0)+φ(0)ψ(1)+φ(1)ψ(0)]+O(θ2), splitp_θ^*(t,x)&= ^(0)ψ^(0)+θ [ ^(0)ψ^(0) ψ^(0)\\ & + ^(0)ψ^(1)+ ^(1)ψ^(0) ]+O(θ^2), split (32) and, writing p0:=φ(0)ψ(0)p_0:= ^(0)ψ^(0) and χ:=ψ(1)/ψ(0)χ:=ψ^(1)/ψ^(0), the correction p1p_1 satisfies the linearised Fokker–Planck equation ∂tp1=−∇⋅((f+κt∇logψ(0))p1)+κt2Δp1−κt∇⋅((∇logψ(0)+∇χ)p0), split _tp_1&=-∇· ( (f+ _t∇ ψ^(0) )p_1 )+ _t2 p_1\\ & - _t∇· ( (∇ ψ^(0)+∇χ )p_0 ), split (33) with ∫p1(t,x)x=0 p_1(t,x)\,dx=0 for all t. Proof. With ∇logψθ=∇logψ(0)+θ∇χ+O(θ2)∇ _θ=∇ ψ^(0)+θ∇χ+O(θ^2), matching the O(θ)O(θ) terms of the exact Fokker–Planck equation gives (33); the integral constraint follows by differentiating ∫pθ∗x=1 p_θ^*\,dx=1 at θ=0θ=0. ∎ The first-order effect of risk sensitivity is therefore exactly localized: forward mass is depleted where the score energy ℐ[ψ(0)]:=‖∇logψ(0)‖2I[ψ^(0)]:=\|∇ ψ^(0)\|^2 of the standard bridge is large (near saddles, ridges, or the ambiguous zone between modes of a mixture target), so the ensemble commits to a basin earlier and crosses transition zones faster: a design handle on which trajectories a generative sampler avoids. VII Conclusion We have constructed a risk-sensitive deformation of the Schrödinger bridge that preserves exact terminal matching and the linearising Cole–Hopf structure, and proved that its value admits no fixed-reference KL dual: the obstruction is an explicit θ/2θ/2 heat-equation defect, ruling out every endpoint-regular reference; per-θ representations are circular, and no single reference serves two distinct θ. Open problems include a general existence theory, contraction of the fixed-point iteration for θ>0θ>0, the limit θ→1θ→ 1, and a duality theory for θJ_θ. VIII Acknowledgments The author used large language models to assist with editing, code refactoring, and the organization of computational experiments. All mathematical derivations, scientific claims, and final code were independently checked, validated, and approved by the author. References [1] C. Léonard (2014) A survey of the Schrödinger problem and some of its connections with optimal transport. 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