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January 2016 A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations
Joaquin Fontbona, Benjamin Jourdain
Ann. Probab. 44(1): 131-170 (January 2016). DOI: 10.1214/14-AOP969

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.

Citation

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Joaquin Fontbona. Benjamin Jourdain. "A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations." Ann. Probab. 44 (1) 131 - 170, January 2016. https://doi.org/10.1214/14-AOP969

Information

Received: 1 July 2013; Revised: 1 February 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1351.60070
MathSciNet: MR3456334
Digital Object Identifier: 10.1214/14-AOP969

Subjects:
Primary: 26D10, 35B40, 37A35, 60H10, 60H30

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 1 • January 2016
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