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Time reversal of diffusion processes under a finite entropy condition

Patrick Cattiaux, Giovanni Conforti, Ivan Gentil, Christian Léonard

TL;DR

The paper develops a unified, entropy-driven framework for time reversal of diffusion processes by deriving an integration by parts formula for the carré du champ using Nelson's stochastic derivatives. Under a finite entropy condition $H(P|R)<\infty$, the time-reversed process $P^*$ is characterized by explicit drift corrections in both Euclidean and abstract settings, accommodating drifts with low regularity. It further decomposes entropic interpolations into current and osmotic components, connecting Schrödinger-type problems to Benamou–Brenier transport via a Fisher-information–based osmotic term. The results extend to random walks on graphs, illustrating robustness beyond diffusions and highlighting links to entropic optimal transport and functional inequalities. Overall, the work provides a broad, rigorous method to analyze time reversal under minimal regularity assumptions, with implications for Schrödinger problems and related transport theories.

Abstract

Motivated by entropic optimal transport, time reversal of diffusion processes is revisited. An integration by parts formula is derived for the carré du champ of a Markov process in an abstract space. It leads to a time reversal formula for a wide class of diffusion processes in $ \mathbb{R}^n$ possibly with singular drifts, extending the already known results in this domain. The proof of the integration by parts formula relies on stochastic derivatives. Then, this formula is applied to compute the semimartingale characteristics of the time-reversed $P^*$ of a diffusion measure $P$ provided that the relative entropy of $P$ with respect to another diffusion measure $R$ is finite, and the semimartingale characteristics of the time-reversed $R^*$ are known (for instance when the reference path measure $R$ is reversible). As an illustration of the robustness of this method, the integration by parts formula is also employed to derive a time-reversal formula for a random walk on a graph.

Time reversal of diffusion processes under a finite entropy condition

TL;DR

The paper develops a unified, entropy-driven framework for time reversal of diffusion processes by deriving an integration by parts formula for the carré du champ using Nelson's stochastic derivatives. Under a finite entropy condition , the time-reversed process is characterized by explicit drift corrections in both Euclidean and abstract settings, accommodating drifts with low regularity. It further decomposes entropic interpolations into current and osmotic components, connecting Schrödinger-type problems to Benamou–Brenier transport via a Fisher-information–based osmotic term. The results extend to random walks on graphs, illustrating robustness beyond diffusions and highlighting links to entropic optimal transport and functional inequalities. Overall, the work provides a broad, rigorous method to analyze time reversal under minimal regularity assumptions, with implications for Schrödinger problems and related transport theories.

Abstract

Motivated by entropic optimal transport, time reversal of diffusion processes is revisited. An integration by parts formula is derived for the carré du champ of a Markov process in an abstract space. It leads to a time reversal formula for a wide class of diffusion processes in possibly with singular drifts, extending the already known results in this domain. The proof of the integration by parts formula relies on stochastic derivatives. Then, this formula is applied to compute the semimartingale characteristics of the time-reversed of a diffusion measure provided that the relative entropy of with respect to another diffusion measure is finite, and the semimartingale characteristics of the time-reversed are known (for instance when the reference path measure is reversible). As an illustration of the robustness of this method, the integration by parts formula is also employed to derive a time-reversal formula for a random walk on a graph.

Paper Structure

This paper contains 9 sections, 26 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Stochastic derivatives
  3. Integration by parts formula
  4. Time reversal of a diffusion process in $\mathbb{R}^n$
  5. Time reversal of a diffusion process. Abstract setting
  6. Current-osmosis decomposition
  7. Random walks
  8. Stochastic derivatives and extended generators
  9. Relative entropy with respect to an unbounded measure

Key Result

Theorem 1.12

Under the Hypotheses ass-01 on $R$ given at eq-09, let $P\in\mathrm{P}(\Omega)$ be Markov and such that Then, for all $t$ the density $\mu_t:=dP_t/d\mathrm{Leb}$ exists and the time reversal $P^*$ of $P$ is a solution of the martingale problem with where the divergence is in the sense of distributions. This is an extension of eq-37 to a low regularity setting which is made precise as follows. F

Theorems & Definitions (61)

  • Theorem 1.12: Time reversal formula
  • Theorem 1.16: Time-reversal formula, again
  • proof
  • Remark 1.18
  • Remark 3.4
  • Lemma 3.6
  • proof
  • Lemma 3.9
  • proof
  • Corollary 3.13
  • ...and 51 more