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

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 (not 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 Itô’s calculus and ideas of Arnold, Carlen and Ju, we obtain moreover a new Bakry–Emery criterion which ensures exponential convergence of the entropy to $0$. This criterion is nonintrinsic 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 nonintrinsic criterion applies without modifying the law of the diffusion process.