Time reversal of diffusion processes under a finite entropy condition

TL;DR

This paper develops a general framework for time reversal of diffusion processes with finite entropy, extending existing results to cases with singular drifts and applying the approach to Markov processes and random walks.

Contribution

It introduces a new integration by parts formula for the carré du champ of Markov processes, enabling broader time reversal results under finite entropy conditions.

Findings

Derived a time reversal formula for diffusions with singular drifts.

Extended time reversal results to processes with finite relative entropy.

Applied the method to random walks on graphs.

Abstract

Motivated by entropic optimal transport, time reversal of diffusion processes is revisited. An integration by parts formula is derived for the carr\'e 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.