Convex entropy decay via the Bochner–Bakry–Emery approach

Pietro Caputo; Paolo Dai Pra; Gustavo Posta

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 734-753
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Caputo, Pietro, Dai Pra, Paolo, and Posta, Gustavo. "Convex entropy decay via the Bochner–Bakry–Emery approach." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 734-753. <http://eudml.org/doc/78041>.

@article{Caputo2009,
abstract = {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.},
author = {Caputo, Pietro, Dai Pra, Paolo, Posta, Gustavo},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {entropy decay; modified logarithmic Sobolev inequality; stochastic particle systems},
language = {eng},
number = {3},
pages = {734-753},
publisher = {Gauthier-Villars},
title = {Convex entropy decay via the Bochner–Bakry–Emery approach},
url = {http://eudml.org/doc/78041},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 734
EP - 753
AB - 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.
LA - eng
KW - entropy decay; modified logarithmic Sobolev inequality; stochastic particle systems
UR - http://eudml.org/doc/78041
ER -

References

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  1. [1] D. Bakry and M. Émery. Diffusions hypercontractives. In Séminaire de Probabilités XIX 177–206. Lecture Notes in Math. 1123. Springer, Berlin, 1985. Zbl0561.60080MR889476
  2. [2] S. G. Bobkov and M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347–365. Zbl0920.60002MR1636948
  3. [3] S. G. Bobkov and P. Tetali. Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 (2006) 289–336. Zbl1113.60072MR2283379
  4. [4] S. Bochner. Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 (1946) 776–797. Zbl0060.38301MR18022
  5. [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. Zbl1087.60071MR2200172
  6. [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. Zbl1072.82001MR2073330
  7. [7] P. Caputo and G. Posta. Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields 139 (2007) 65–87. Zbl1126.60082MR2322692
  8. [8] P. Caputo and P. Tetali. Unpublished notes, 2005. 
  9. [9] D. Chafai. Binomial–Poisson entropic inequalities and the M/M/∞ queue. ESAIM Probab. Stat. 10 (2006) 317–339. Zbl1188.60047MR2247924
  10. [10] P. Dai Pra, A. M. Paganoni and G. Posta. Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002) 1959–1976. Zbl1013.60076MR1944012
  11. [11] P. Dai Pra and G. Posta. Logarithmic Sobolev inequality for zero-range dynamics. Ann. Probab. 33 (2005) 2355–2401. Zbl1099.60068MR2184099
  12. [12] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695–750. Zbl0867.60043MR1410112
  13. [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. Zbl1046.60003MR2023890
  14. [14] S. Goel. Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51–79. Zbl1074.60080MR2094147
  15. [15] O. Johnson. Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 (2007) 791–802. Zbl1115.60012MR2327839
  16. [16] A. Joulin. Poisson-type deviation inequalities for curved continuous-time Markov chains. Bernoulli 13 (2007) 782–798. Zbl1131.60069MR2348750
  17. [17] C. Landim, S. Sethuraman and S. R. S. Varadhan. Spectral gap for zero-range dynamics. Ann. Probab. 24 (1996) 1871–1902. Zbl0870.60095MR1415232
  18. [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. Zbl0979.60096MR1837286
  19. [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. Zbl0793.60111MR1269388
  20. [20] L. Miclo. An example of application of discrete Hardy’s inequalities. Markov Process. Related Fields 5 (1999) 319–330. Zbl0942.60081MR1710983
  21. [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. Zbl0758.60070MR1182416
  22. [22] L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000) 427–438. Zbl0970.60093MR1800540
  23. [23] H.-T. Yau. Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 (1996) 367–408. Zbl0864.60079MR1414837

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