A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations

TL;DR

This paper provides a trajectorial interpretation of entropy and Fisher information dissipation in stochastic differential equations, deriving a stochastic entropy dissipation formula and a new criterion for exponential convergence.

Contribution

It introduces a trajectorial framework for entropy dissipation, derives a stochastic analogue of the entropy formula, and proposes a non-intrisic Bakry-Emery criterion for exponential convergence.

Findings

Derived a stochastic entropy dissipation formula for Markov diffusions.

Established a new non-intrisic Bakry-Emery criterion for exponential convergence.

Provided examples where classic criteria fail but the new criterion applies.

Abstract

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit the Doob-Meyer decomposition of this submartingale. We deduce a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptions, and using It\^o's calculus and ideas of Arnold, Carlen and Ju \cite{Arnoldcarlenju}, we obtain moreover a new Bakry Emery criterion which ensures exponential convergence of the entropy to $0$. This criterion is non-intrisic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. We provide examples where the classic Bakry Emery criterion fails, but our non-intrisic criterion applies without modifying the law of the diffusion process.