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.
Mots clés : entropy decay, modified logarithmic Sobolev inequality, stochastic particle systems
@article{AIHPB_2009__45_3_734_0, author = {Caputo, Pietro and Dai Pra, Paolo and Posta, Gustavo}, title = {Convex entropy decay via the {Bochner-Bakry-Emery} approach}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {734--753}, publisher = {Gauthier-Villars}, volume = {45}, number = {3}, year = {2009}, doi = {10.1214/08-AIHP183}, mrnumber = {2548501}, zbl = {1181.60142}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP183/} }
TY - JOUR AU - Caputo, Pietro AU - Dai Pra, Paolo AU - Posta, Gustavo TI - Convex entropy decay via the Bochner-Bakry-Emery approach JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 734 EP - 753 VL - 45 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP183/ DO - 10.1214/08-AIHP183 LA - en ID - AIHPB_2009__45_3_734_0 ER -
%0 Journal Article %A Caputo, Pietro %A Dai Pra, Paolo %A Posta, Gustavo %T Convex entropy decay via the Bochner-Bakry-Emery approach %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 734-753 %V 45 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP183/ %R 10.1214/08-AIHP183 %G en %F AIHPB_2009__45_3_734_0
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, pp. 734-753. doi : 10.1214/08-AIHP183. http://www.numdam.org/articles/10.1214/08-AIHP183/
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