no. 3
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.

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.

DOI : 10.1214/08-AIHP183
Classification : 39B62, 60J80, 60K35
Mots clés : entropy decay, modified logarithmic Sobolev inequality, stochastic particle systems 
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     title = {Convex entropy decay via the {Bochner-Bakry-Emery} approach},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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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] D. Bakry and M. Émery. Diffusions hypercontractives. In Séminaire de Probabilités XIX 177-206. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | Numdam | MR | Zbl

[2] S. G. Bobkov and M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347-365. | MR | Zbl

[3] S. G. Bobkov and P. Tetali. Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 (2006) 289-336. | MR | Zbl

[4] S. Bochner. Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 (1946) 776-797. | MR | Zbl

[5] A. S. Boudou, P. Caputo, P. Dai Pra and G. Posta. Spectral gap estimates for interacting particle systems via a Bochner-type identity. J. Funct. Anal. 232 (2006) 222-258. | MR | Zbl

[6] P. Caputo. 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] P. Caputo and G. Posta. Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields 139 (2007) 65-87. | MR | Zbl

[8] P. Caputo and P. Tetali. Unpublished notes, 2005.

[9] D. Chafai. Binomial-Poisson entropic inequalities and the M/M/∞ queue. ESAIM Probab. Stat. 10 (2006) 317-339. | Numdam | MR

[10] P. Dai Pra, A. M. Paganoni and G. Posta. Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002) 1959-1976. | MR | Zbl

[11] P. Dai Pra and G. Posta. Logarithmic Sobolev inequality for zero-range dynamics. Ann. Probab. 33 (2005) 2355-2401. | MR | Zbl

[12] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695-750. | MR | Zbl

[13] F. Gao and J. Quastel. Exponential decay of entropy in the random transposition and Bernoulli-Laplace models. Ann. Appl. Probab. 13 (2003) 1591-1600. | MR | Zbl

[14] S. Goel. Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51-79. | MR | Zbl

[15] O. Johnson. Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 (2007) 791-802. | MR | Zbl

[16] A. Joulin. Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 (2007) 782-798. | MR | Zbl

[17] C. Landim, S. Sethuraman and S. R. S. Varadhan. Spectral gap for zero-range dynamics. Ann. Probab. 24 (1996) 1871-1902. | MR | Zbl

[18] M. Ledoux. 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] F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Comm. Math. Phys. 161 (1994) 487-514. | MR | Zbl

[20] L. Miclo. An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | MR | Zbl

[21] D. W. Stroock and B. Zegarliński. The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 (1992) 175-193. | MR | Zbl

[22] L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000) 427-438. | MR | Zbl

[23] H.-T. Yau. Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 (1996) 367-408. | MR | Zbl

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