Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré

Journal of Functional Analysis

Volumn 254, Issue 3, 2008, Pages 727-759

  Bakry, Dominique ^a,b   Cattiaux, Patrick ^a,c   Guillin, Arnaud ^d,e  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b INSTITUT UNIVERSITAIRE DE FRANCE   (France)

^c CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^d Ecole Centrale Marseille   (France)

^e AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Ergodic processes; Hypocoercivity; Lyapunov functions; Poincar inequalities

Indexed keywords

EID: 37249083912     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2007.11.002     Document Type: Article

References (37)

1. Ané C., Blachère S., Chafaï D., Fougères P., Gentil I., Malrieu F., Roberto C., and Scheffer G. Sur les inégalités de Sobolev logarithmiques. Panor. Synthèses vol. 10 (2000), Soc. Math. France, Paris
2. Bakry D. L'hypercontractivité et son utilisation en théorie des semigroupes. Lectures on Probability Theory, École d'été de Probabilités de St-Flour 1992. Lecture Notes in Math. vol. 1581 (1994), Springer, Berlin 1-114
3. Barthe F., Cattiaux P., and Roberto C. Concentration for independent random variables with heavy tails. AMRX Appl. Math. Res. Express. 2005 2 (2005) 39-60
4. Barthe F., Cattiaux P., and Roberto C. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoamericana 22 (2006) 993-1066
5. Barthe F., Cattiaux P., and Roberto C. Isoperimetry between exponential and Gaussian. Electron. J. Probab. 12 (2007) 1212-1237
6. Cattiaux P. A pathwise approach of some classical inequalities. Potential Anal. 20 (2004) 361-394
7. Cattiaux P. Hypercontractivity for perturbed diffusion semi-groups. Ann. Fac. Sci. Toulouse Mat. 14 4 (2005) 609-628
8. Cattiaux P., Gentil I., and Guillin A. Weak logarithmic Sobolev inequalities and entropic convergence. Probab. Theory Related Fields 139 3-4 (2007) 563-603
9. Douc R., Moulines E., and Rosenthal J.S. Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Probab. 14 4 (2004) 1643-1665
10. Douc R., Fort G., and Guillin A. Subgeometric rates of convergence of f-ergodic strong Markov processes. preprint, available at (2006). arXiv: math.ST/0605791
11. Down N., Meyn S.P., and Tweedie R.L. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 4 (1995) 1671-1691
12. Fort G., and Roberts G.O. Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15 2 (2005) 1565-1589
13. Franchi B. Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. Trans Amer. Math. Soc. 327 1 (1991) 125-158
14. Gentil I. Inégalités de Sobolev logarithmiques et de Poincaré pour la loi uniforme. preprint, available at (2006). http://www.ceremade.dauphine.fr/gentil/logsob-segment.pdf
15. Gong F.Z., and Wu L.M. Spectral gap of positive operators and its applications. C. R. Acad. Sci. Paris Sér. I 331 (2000) 983-988
16. Hairer M., and Mattingly J.C. Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations. preprint, available at (2006). arXiv: math.PR/0602479
17. Helffer B., and Nier F. Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Lectures Notes in Math. vol. 1862 (2005), Springer, Berlin
18. Hérau F., and Nier F. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with high degree potential. Arch. Ration. Mech. Anal. 171 2 (2004) 151-218
19. Jerison D. The Poincaré inequality for vector fields satisfying Hörmander's condition. Duke Math. J. 53 (1986) 503-523
20. Lu G. Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications. Rev. Mat. Iberoamericana 8 3 (1992) 367-439
21. Meyn S.P., and Tweedie R.L. Markov Chains and Stochastic Stability. Commun. Control Engrg. Ser. (1993), Springer, London
22. Meyn S.P., and Tweedie R.L. Stability of Markovian processes II: Continuous-time processes and sampled chains. Adv. Appl. Probab. 25 (1993) 487-517
23. Meyn S.P., and Tweedie R.L. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25 (1993) 518-548
24. Roberto C., and Zegarlinski B. Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups. J. Funct. Anal. 243 1 (2007) 28-66
25. 2-convergence rates of Markov semigroups. J. Funct. Anal. 185 2 (2001) 564-603
26. Royer G. Une initiation aux inégalités de Sobolev logarithmiques (1999), Soc. Math. France, Paris
27. Saloff-Coste L. On global Sobolev inequalities. Forum Math. 6 (1994) 271-286
28. Veretennikov A.Yu. On polynomial mixing bounds for stochastic differential equations. Stoch. Process. Appl. 70 1 (1997) 115-127
29. C. Villani, Hypocoercive diffusion operators, preprint, Proceedings of the International Congress of Mathematicians, Madrid 2006, available on author's Web page, 2006
30. C. Villani, Hypocoercive diffusion operators in Hörmander form, preprint, available on author's Web page, 2006
31. C. Villani, Hypocoercivity, preprint, available on author's Web page, 2006
32. Wang F.Y. Functional inequalities for empty essential spectrum. J. Funct. Anal. 170 1 (2000) 219-245
33. Wang F.Y. A generalization of Poincaré and log-Sobolev inequalities. Potential Anal. 22 1 (2005) 1-15
34. Wu L. Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172 2 (2000) 301-376
35. Wu L. Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stoch. Process. Appl. 91 (2001) 205-238
36. Wu L. Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Related Fields 128 2 (2004) 255-321
37. Zitt P.A. Annealing diffusion in a slowly growing potential. Stoch. Process. Appl. 118 1 (2008) 76-119