Tuesday 26th June 2012 – 14:15 to 15:15
Speaker(s): Joaquin Fontbona (Center for Mathematical Modeling, University of Chile)
We develop a pathwise description of the dissipation of general convex entropies for continuous time Markov processes, based on martingales and convergence theorems with respect to the tail sigma field. The entropy is in this setting the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit its Doob-Meyer decomposition, thereby providing a stochastic analogue of the well known entropy dissipation formula, valid for general convex entropies (including total variation). Under additional regularity assumptions, and using Itô calculus and some ideas of Arnold, Carlen and Ju, we obtain a new Bakry Emery criterion which ensures exponential convergence of the entropy to 0. This criterion is non-intrinsic in the sense that it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. We provide a simple example where the classic Bakry Emery criterion fails, but our non-intrinsic criterion ensuring exponential convergence to equilibrium applies. Joint work with Benjamin Jourdain (Cermics ENPC, Paris).
Part of the Stochastic Analysis Seminar Series