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A survey of the Schrödinger problem and some of its connections with optimal transport

Christian Léonard

TL;DR

This survey presents the Schrödinger problem as an entropy-minimization bridge between stochastic path measures and optimal transport. It develops the dynamic and static formulations, the crucial $(f,g)$-transform representation, and standard examples to illustrate the theory. The text also clarifies asymptotic connections to Monge–Kantorovich OT via a slow-down limit and situates the framework within statistical physics through large deviations and the Sanov perspective. Finally, it offers a historical overview and highlights links to stochastic control, reciprocal processes, and curvature-based transport theory, emphasizing both foundational results and newer insights.

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.

A survey of the Schrödinger problem and some of its connections with optimal transport

TL;DR

This survey presents the Schrödinger problem as an entropy-minimization bridge between stochastic path measures and optimal transport. It develops the dynamic and static formulations, the crucial -transform representation, and standard examples to illustrate the theory. The text also clarifies asymptotic connections to Monge–Kantorovich OT via a slow-down limit and situates the framework within statistical physics through large deviations and the Sanov perspective. Finally, it offers a historical overview and highlights links to stochastic control, reciprocal processes, and curvature-based transport theory, emphasizing both foundational results and newer insights.

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.

Paper Structure

This paper contains 8 sections, 14 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Schrödinger's problem
  3. $(f,g)$-transform of a Markov measure
  4. Standard examples
  5. Slowing down
  6. The statistical physics motivation of Schrödinger's problem
  7. A short history of Schrödinger's problem and related literature
  8. Relative entropy with respect to an unbounded measure

Key Result

Proposition 2.3

The Schrödinger problems Sdyn and S admit respectively at most one solution $\widehat{P}\in\mathrm{P}(\Omega)$ and $\widehat{\pi}\in\mathrm{P}(\mathcal{X}^2).$ If Sdyn admits the solution $\widehat{P},$ then $\widehat{\pi}=\widehat{P} _{01}$ is the solution of S. Conversely, if $\widehat{\pi}$ solve which means that and that $\widehat{P}$ shares its bridges with $R:$

Theorems & Definitions (32)

  • Definition 2.1: Dynamic Schrödinger's problem
  • Definition 2.2: Schrödinger's problem
  • Proposition 2.3: Föllmer, Foe85
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • ...and 22 more