Caputo, Pietro; Dai Pra, Paolo; Posta, Gustavo
Convex entropy decay via the Bochner-Bakry-Emery approach
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3 , p. 734-753
Zbl 1181.60142 | MR 2548501 | 4 citations dans Numdam
doi : 10.1214/08-AIHP183
URL stable : http://www.numdam.org/item?id=AIHPB_2009__45_3_734_0

Classification:  39B62,  60J80,  60K35
Nous développons une méthode, insiprée par une identité de Bochner, pour obtenir des estimées sur la decroissance exponentielle de l'entropie relative de processus de Markov avec sauts. Lorsque nous pouvons appliquer cette méthode, l'entropie relative est une fonction convexe du temps. On montre que la méthode s'applique de facon efficace à une large classe de processus de naissance et mort. On considère aussi d'autres exemples, comme les processus de zero-range et de Bernoulli-Laplace dans des cas non-homogènes. Pour ces derniers modèles les résultats connus, obtenus par la méthode de martingale, étaient limités au cas homogène.
We develop a method, based on a Bochner-type identity, to obtain estimates on the exponential rate of decay of the relative entropy from equilibrium of Markov processes in discrete settings. When this method applies the relative entropy decays in a convex way. The method is shown to be rather powerful when applied to a class of birth and death processes. We then consider other examples, including inhomogeneous zero-range processes and Bernoulli-Laplace models. For these two models, known results were limited to the homogeneous case, and obtained via the martingale approach, whose applicability to inhomogeneous models is still unclear.

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