A Survey of the Schrödinger Problem and Some of Its Connections with Optimal Transport

Abstract

This article is aimed at presenting the Schrödinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user’s guide to Schrödinger problem. We also give a survey of the related literature. In addition, some new results are proved.

Summary

Léonard provides the definitive mathematical survey connecting Schrödinger’s 1931 entropy minimization problem to modern optimal transport theory. The paper formulates both the dynamic Schrödinger problem $(S_{\text{dyn}})$ — minimizing $H(P|R)$ over path measures $P$ with prescribed initial and terminal marginals ${\mu }_0,{\mu }_1$ — and the static Schrödinger problem $(S)$ — minimizing $H(\pi |R_{01})$ over couplings $\pi$ with the same marginals. These are shown to be equivalent: inf $(S_{\text{dyn}})$ = inf $(S)$, and the dynamic solution $\hat{P}$ disintegrates via the reference bridges $R^{xy}$ weighted by the static solution $\hat{\pi }$.

The central connection to optimal transport is established through the Γ-convergence of a sequence of “slowed down” Schrödinger problems to the Monge-Kantorovich problem. By replacing the reference measure $R$ with $R^k$ (generator $L^k=L/k$), the rescaled entropy $H(P|R^k)/{\alpha }_k\to \int CdP$, where $C$ is the kinetic action. This proves that the Schrödinger bridge problem is a regular (entropic) approximation of optimal transport, with the static version converging to the Monge-Kantorovich coupling and the dynamic version converging to the displacement interpolation.

The paper also introduces the (f,g)-transform as the time-symmetric generalization of the Doob h-transform, proves Born’s formula $P_t=f_t{g}_{t}m$ for the marginal densities, derives the Schrödinger potentials $\phi =\log f$, $\psi =\log g$ as second-order analogues of Kantorovich potentials, and establishes the entropic interpolation as the SB analogue of displacement interpolation. The statistical physics motivation via Sanov’s theorem and the “lazy gas experiment” provides physical intuition for why entropy minimization selects the most likely particle dynamics.

Key Contributions

• Rigorous formulation of dynamic and static Schrödinger problems in full generality (unbounded reference measures, Polish state spaces)
• Proof that the SB solution has product-shaped Radon-Nikodym derivative $\hat{P}=f_0(X_0)g_1(X_1)R$ (Theorem 2.9/2.12) and is Markov when $R$ is Markov
• Introduction of the (f,g)-transform as the natural generalization of Doob’s h-transform to the time-symmetric (two-boundary) setting
• Derivation of forward/backward generators and Schrödinger potentials solving second-order Hamilton-Jacobi-Bellman equations
• Proof that the Monge-Kantorovich problem is the Γ-limit of Schrödinger problems as diffusivity → 0 (Statement 5.1, Theorem 5.2)
• New Benamou-Brenier type formulas (Propositions 4.1, 4.2) for both continuous diffusions and discrete random walks
• Statistical physics derivation via Sanov’s theorem connecting the SB to the most likely evolution of a large particle system

Methodology

The paper employs relative entropy (KL divergence) on path space as the central variational tool. The reference measure $R$ is allowed to be unbounded (σ-finite), which is necessary for handling the reversible Brownian motion on ${\mathbb{R}}^n$. The key structural results are proved via:

1. The additive property of relative entropy (Appendix A) to decompose $H(P|R)$ into endpoint and bridge components
2. Duality theory: the dual problem $(D)$ involves maximizing over Schrödinger potentials $\phi ,\psi$
3. Γ-convergence via the Laplace-Varadhan integral lemma to pass from entropy to kinetic action
4. Stochastic calculus (forward/backward stochastic derivatives in the sense of Nelson) to derive the dynamics of the $(f,g)$-transform

Key Findings

• The static SB problem is structurally parallel to Monge-Kantorovich: both minimize over couplings with fixed marginals, but SB uses entropy while MK uses cost
• Since $R_{01}(dxdy)\propto \exp (-d(x,y{)}^2/2)\text{vol}(dx)\text{vol}(dy)$, the SB is specifically connected to quadratic optimal transport with cost $c(x,y)=d(x,y{)}^2/2$
• The entropic interpolation $[{\mu }_0,{\mu }_1{]}^k$ is always positive and regular on $(0,1)\times \mathcal{X}$, unlike the displacement interpolation which may concentrate on thin sets
• As $k\to \infty$: entropic interpolation → displacement interpolation, Schrödinger potentials → Kantorovich potentials, SB bridges → geodesic paths
• The entropy along entropic interpolations satisfies a rigorous second-derivative formula (eq. 6.7), connecting to Ricci curvature bounds via Bochner’s formula — the starting point of Lott-Sturm-Villani theory

Important References

1. Optimal Transport Old and New — Villani’s comprehensive OT textbook, providing the Monge-Kantorovich framework underlying the convergence results
2. From the Schrödinger Problem to the Monge-Kantorovich Problem — Léonard’s companion paper proving the rigorous Γ-convergence results for continuous diffusions
3. Random Fields and Diffusion Processes — Föllmer’s Saint-Flour lectures that first explicitly formulated the Schrödinger problem as entropy minimization

Atomic Notes

• (f,g)-transform
• Schrödinger system
• Schrödinger potentials
• entropic interpolation
• displacement interpolation
• Γ-convergence of Schrödinger to Monge-Kantorovich
• Benamou-Brenier formula

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