Time reversal of diffusion processes under a finite entropy condition

Patrick Cattiaux (1) , Giovanni Conforti (2) , Ivan Gentil (3) , Christian Léonard (4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 ICJ - Institut Camille Jordan
4 MODAL'X - Modélisation aléatoire de Paris X

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 Rn 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.

Mots clés: April; 2021. 2010 Mathematics Subject Classification. 60J60; 60J25 Time-reversal; diffusion process; stochastic derivative; relative entropy; random walk; entropic optimal transport

Domaines: Probabilités [math.PR]

Patrick Cattiaux, Giovanni Conforti, Ivan Gentil, Christian Léonard. Time reversal of diffusion processes under a finite entropy condition. 2021. ⟨hal-03206478v1⟩