Convex entropy decay via the Bochner-Bakry-Emery approach
Caputo, Pietro ; Dai Pra, Paolo ; Posta, Gustavo
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009), p. 734-753 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2009-01-01
DOI : https://doi.org/10.1214/08-AIHP183
Classification:  39B62,  60J80,  60K35 
@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},
     volume = {45},
     year = {2009},
     pages = {734-753},
     doi = {10.1214/08-AIHP183},
     mrnumber = {2548501},
     zbl = {1181.60142},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 734-753. doi : 10.1214/08-AIHP183. http://gdmltest.u-ga.fr/item/AIHPB_2009__45_3_734_0/

[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 889476 | Zbl 0561.60080

[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 1636948 | Zbl 0920.60002

[3] S. G. Bobkov and P. Tetali. Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 (2006) 289-336. | MR 2283379 | Zbl 1113.60072

[4] S. Bochner. Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 (1946) 776-797. | MR 18022 | Zbl 0060.38301

[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 2200172 | Zbl 1087.60071

[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 2073330 | Zbl 1072.82001

[7] P. Caputo and G. Posta. Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields 139 (2007) 65-87. | MR 2322692 | Zbl 1126.60082

[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 2247924 | Zbl pre05216861

[10] P. Dai Pra, A. M. Paganoni and G. Posta. Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002) 1959-1976. | MR 1944012 | Zbl 1013.60076

[11] P. Dai Pra and G. Posta. Logarithmic Sobolev inequality for zero-range dynamics. Ann. Probab. 33 (2005) 2355-2401. | MR 2184099 | Zbl 1099.60068

[12] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695-750. | MR 1410112 | Zbl 0867.60043

[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 2023890 | Zbl 1046.60003

[14] S. Goel. Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51-79. | MR 2094147 | Zbl 1074.60080

[15] O. Johnson. Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 (2007) 791-802. | MR 2327839 | Zbl 1115.60012

[16] A. Joulin. Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 (2007) 782-798. | MR 2348750 | Zbl 1131.60069

[17] C. Landim, S. Sethuraman and S. R. S. Varadhan. Spectral gap for zero-range dynamics. Ann. Probab. 24 (1996) 1871-1902. | MR 1415232 | Zbl 0870.60095

[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. | Numdam | MR 1837286 | Zbl 0979.60096

[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 1269388 | Zbl 0793.60111

[20] L. Miclo. An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | MR 1710983 | Zbl 0942.60081

[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 1182416 | Zbl 0758.60070

[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 1800540 | Zbl 0970.60093

[23] H.-T. Yau. Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 (1996) 367-408. | MR 1414837 | Zbl 0864.60079