Time reversal of Markov processes with jumps under a finite entropy condition

TL;DR

This paper investigates the time reversal of Markov jump processes in Euclidean space using entropic methods, providing a general framework for processes with complex jump behaviors and unbounded variation.

Contribution

It introduces a novel approach to compute the semimartingale characteristics of time-reversed Markov jump processes under finite entropy conditions, extending previous methods to more general jump processes.

Findings

Derived explicit formulas for time-reversed process characteristics.

Extended the class of jump processes for which time reversal can be analyzed.

Connected entropic optimal transport with stochastic process time reversal.

Abstract

Motivated by entropic optimal transport, time reversal of Markov jump processes in $\mathbb{R}^n$ is investigated. Relying on an abstract integration by parts formula for the carr\'e du champ of a Markov process recently obtained by Cattiaux, Gentil and the authors, and using an entropic improvement strategy discovered by F\"ollmer in the eighties, we compute the semimartingale characteristics of the time reversed process for a wide class of jump processes with possibly unbounded variation sample paths and singular intensities of jump.