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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1214/08-AIHP183/
[1] Diffusions hypercontractives. In Séminaire de Probabilités XIX 177-206. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | Numdam | MR | Zbl
and .[2] On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347-365. | MR | Zbl
and .[3] Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 (2006) 289-336. | MR | Zbl
and .[4] Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 (1946) 776-797. | MR | Zbl
.[5] Spectral gap estimates for interacting particle systems via a Bochner-type identity. J. Funct. Anal. 232 (2006) 222-258. | MR | Zbl
, , and .[6] Spectral gap inequalities in product spaces with conservation laws. In Advanced Studies in Pure Mathematics 39. H. Osada and T. Funaki (Eds). Math. Soc. Japan, Tokyo, 2004. | MR | Zbl
.[7] Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields 139 (2007) 65-87. | MR | Zbl
and .[8] Unpublished notes, 2005.
and .[9] Binomial-Poisson entropic inequalities and the M/M/∞ queue. ESAIM Probab. Stat. 10 (2006) 317-339. | Numdam | MR
.[10] Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002) 1959-1976. | MR | Zbl
, and .[11] Logarithmic Sobolev inequality for zero-range dynamics. Ann. Probab. 33 (2005) 2355-2401. | MR | Zbl
and .[12] Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695-750. | MR | Zbl
and .[13] Exponential decay of entropy in the random transposition and Bernoulli-Laplace models. Ann. Appl. Probab. 13 (2003) 1591-1600. | MR | Zbl
and .[14] Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51-79. | MR | Zbl
.[15] Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 (2007) 791-802. | MR | Zbl
.[16] Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 (2007) 782-798. | MR | Zbl
.[17] Spectral gap for zero-range dynamics. Ann. Probab. 24 (1996) 1871-1902. | MR | Zbl
, and .[18] Logarithmic Sobolev inequalities for unbounded spin systems revisited. In Séminaire de Probabilités XXXV 167-194. Lecture Notes in Math. 1755. Springer, Berlin, 2001. | EuDML | Numdam | MR | Zbl
.[19] Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Comm. Math. Phys. 161 (1994) 487-514. | MR | Zbl
and .[20] An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | MR | Zbl
.[21] The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 (1992) 175-193. | MR | Zbl
and .[22] A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000) 427-438. | MR | Zbl
.[23] Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 (1996) 367-408. | MR | Zbl
.Cité par Sources :