Trends to equilibrium in total variation distance
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 117-145.

Nous étudions ici la vitesse de convergence, pour la distance en variation totale, de diffusions ergodiques dont la loi initiale satisfait une intégrabilité donnée. Nous présentons différentes approches basées sur l’utilisation d’inégalités fonctionnelles. La première étape consiste à donner une borne générale à la Pinsker. Cette borne permet alors d’utiliser, en les combinant à une procedure de troncature, des inégalités usuelles (telles Poincaré ou Poincaré faibles,…). Dans un deuxième temps nous introduisons de nouvelles inégalités appelées ψ que nous caractérisons à l’aide de condition de type capacité-mesure et d’inégalités de type F-Sobolev. Une étude directe de la distance de Hellinger est également proposée. Pour conclure, une approche dynamique basée sur le renversement du rôle du semigroupe de diffusion et de la mesure invariante permet d'obtenir de nouvelles bornes intéressantes.

This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,…) and truncation procedure, and secondly through the introduction of new functional inequalities ψ . These ψ -inequalities are characterized through measure-capacity conditions and F-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds.

DOI : 10.1214/07-AIHP152
Classification : 26D10, 60E15
Mots clés : total variation, diffusion processes, speed of convergence, Poincaré inequality, logarithmic Sobolev inequality, F-Sobolev inequality
@article{AIHPB_2009__45_1_117_0,
     author = {Cattiaux, Patrick and Guillin, Arnaud},
     title = {Trends to equilibrium in total variation distance},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {117--145},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     doi = {10.1214/07-AIHP152},
     mrnumber = {2500231},
     zbl = {1202.26028},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/07-AIHP152/}
}
TY  - JOUR
AU  - Cattiaux, Patrick
AU  - Guillin, Arnaud
TI  - Trends to equilibrium in total variation distance
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 117
EP  - 145
VL  - 45
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/07-AIHP152/
DO  - 10.1214/07-AIHP152
LA  - en
ID  - AIHPB_2009__45_1_117_0
ER  - 
%0 Journal Article
%A Cattiaux, Patrick
%A Guillin, Arnaud
%T Trends to equilibrium in total variation distance
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 117-145
%V 45
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/07-AIHP152/
%R 10.1214/07-AIHP152
%G en
%F AIHPB_2009__45_1_117_0
Cattiaux, Patrick; Guillin, Arnaud. Trends to equilibrium in total variation distance. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 117-145. doi : 10.1214/07-AIHP152. http://archive.numdam.org/articles/10.1214/07-AIHP152/

[1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer. Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses 10. Société Mathématique de France, Paris, 2000. | MR | Zbl

[2] D. Bakry. L'hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on Probability theory. École d'été de Probabilités de St-Flour 1992 1-114. Lecture Notes in Math. 1581. Springer, Berlin, 1994. | MR | Zbl

[3] D. Bakry, P. Cattiaux and A. Guillin. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 (2008) 727-759. | MR | Zbl

[4] D. Bakry, M. Ledoux and F. Y. Wang. Perturbations of inequalities under growth conditions. J. Math. Pures Appl. 87 (2007) 394-407. | MR | Zbl

[5] F. Barthe, P. Cattiaux and C. Roberto. Concentration for independent random variables with heavy tails. AMRX 2005 (2005) 39-60. | MR | Zbl

[6] F. Barthe, P. Cattiaux and C. Roberto. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Math. Iberoamericana 22 (2006) 993-1066. | MR | Zbl

[7] F. Barthe, P. Cattiaux and C. Roberto. Isoperimetry between exponential and Gaussian. Electron. J. Probab. 12 (2007) 1212-1237. | MR | Zbl

[8] F. Barthe and C. Roberto. Sobolev inequalities for probability measures on the real line. Studia Math. 159 (2003) 481-497. | MR | Zbl

[9] L. Bertini and B. Zegarlinski. Coercive inequalities for Gibbs measures. J. Funct. Anal. 162 (1999) 257-286. | MR | Zbl

[10] P. Cattiaux. A pathwise approach of some classical inequalities. Potential Anal. 20 (2004) 361-394. | MR | Zbl

[11] P. Cattiaux. Hypercontractivity for perturbed diffusion semi-groups. Ann. Fac. Sc. Toulouse 14 (2005) 609-628. | Numdam | MR | Zbl

[12] P. Cattiaux, I. Gentil and A. Guillin. Weak logarithmic-Sobolev inequalities and entropic convergence. Probab. Theory Related Fields 139 (2007) 563-603. | MR | Zbl

[13] P. Cattiaux and A. Guillin. Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Statist. 12 (2008) 12-29. | Numdam | MR

[14] P. Cattiaux and A. Guillin. On quadratic transportation cost inequalities. J. Math. Pures Appl. 88 (2006) 341-361. | MR | Zbl

[15] P. Cattiaux and A. Guillin. Trends to equilibrium in total variation distance. Available at ArXiv.math.PR/0703451, 2007. Stochastic Process. Appl. To appear.

[16] E. B. Davies. Heat Kernels and Spectral Theory. Cambridge Univ. Press, 1989. | MR | Zbl

[17] P. Del Moral, M. Ledoux and L. Miclo. On contraction properties of Markov kernels. Probab. Theory Related Fields 126 (2003) 395-420. | MR | Zbl

[18] J. Dolbeault, I. Gentil, A. Guillin and F.Y. Wang. Lq-functional inequalities and weighted porous media equations. Potential Anal. 28 (2008) 35-59. | MR | Zbl

[19] R. Douc, G. Fort and A. Guillin. Subgeometric rates of convergence of f-ergodic strong Markov processes. Preprint. Available at ArXiv.math.ST/0605791, 2006. | MR | Zbl

[20] N. Down, S. P. Meyn and R. L. Tweedie. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 (1995) 1671-1691. | MR | Zbl

[21] G. Fort and G. O. Roberts. Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15 (2005) 1565-1589. | MR | Zbl

[22] O. Kavian, G. Kerkyacharian and B. Roynette. Some remarks on ultracontractivity. J. Funct. Anal. 111 (1993) 155-196. | MR | Zbl

[23] R. Latała and K. Oleszkiewicz. Between Sobolev and Poincaré. In Geometric Aspects of Functional Analysis 147-168. Lecture Notes in Math. 1745. Springer, Berlin, 2000. | MR | Zbl

[24] Y. H. Mao. Strong ergodicity for Markov processes by coupling. J. Appl. Probab. 39 (2002) 839-852. | MR | Zbl

[25] V. G. Maz'Ja. Sobolev Spaces. Springer, Berlin, 1985. (Translated from the Russian by T. O. Shaposhnikova.) | MR | Zbl

[26] S. P. Meyn and R. L. Tweedie. Stability of markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Probab. 25 (1993) 487-517. | MR | Zbl

[27] S. P. Meyn and R. L. Tweedie. Stability of markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25 (1993) 518-548. | MR | Zbl

[28] C. Roberto and B. Zegarlinski. Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups. J. Funct. Anal. 243 (2007) 28-66. | MR | Zbl

[29] M. Röckner and F. Y. Wang. Weak Poincaré inequalities and L2-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001) 564-603. | MR | Zbl

[30] A. Y. Veretennikov. On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 (1997) 115-127. | MR | Zbl

[31] F. Y. Wang. Functional inequalities for empty essential spectrum. J. Funct. Anal. 170 (2000) 219-245. | MR | Zbl

[32] F. Y. Wang. Functional Inequalities, Markov Processes and Spectral Theory. Science Press, Beijing, 2004. | MR

[33] F. Y. Wang. Probability distance inequalities on Riemannian manifolds and path spaces. J. Funct. Anal. 206 (2004) 167-190. | MR | Zbl

[34] F. Y. Wang. A generalization of Poincaré and log-Sobolev inequalities. Potential Anal. 22 (2005) 1-15. | MR | Zbl

[35] F. Y. Wang. L1-convergence and hypercontractivity of diffusion semi-groups on manifolds. Studia Math. 162 (2004) 219-227. | MR | Zbl

[36] P. A. Zitt. Annealing diffusion in a slowly growing potential. Stochastic Process. Appl. 118 (2008) 76-119. | MR | Zbl

Cité par Sources :