Poincaré inequalities and hitting times
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 95-118.

L'équivalence entre le trou spectral, l'intégrabilité exponentielle des temps de retour et des conditions de Lyapunov est bien connue pour les chaînes de Markov. Nous donnons ici cette même équivalence (quantitative) pour des diffusions réversibles. Une des conséquences est la généralisation de résultats de Bobkov dans le cas unidimensionnel sur la valeur de la constante de l'inégalité de Poincaré des mesures log-concaves à des potentiels super linéaires. En conclusion, nous étudions diverses inégalités fonctionnelles sous diffŕentes conditions d'intégrabilité des temps de retour (polynomiale,…). En particulier, en dimension 1, nous montrons l'équivalence entre ultracontractivité et condition de Lyapunov bornée.

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions is well known. We give here the correspondence (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for log-concave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial,…). In particular, in the one dimensional case, ultracontractivity is equivalent to a bounded Lyapunov condition.

DOI : 10.1214/11-AIHP447
Classification : 26D10, 39B62, 47D07, 60G10, 60J60
Mots-clés : poincaré inequalities, Lyapunov functions, hitting times, log-concave measures, Poincaré-Sobolev inequalities 
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Cattiaux, Patrick; Guillin, Arnaud; Zitt, Pierre André. Poincaré inequalities and hitting times. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 95-118. doi : 10.1214/11-AIHP447. http://archive.numdam.org/articles/10.1214/11-AIHP447/

[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. | Zbl

[2] D. Bakry, F. Barthe, P. Cattiaux and A. Guillin. A simple proof of the Poincaré inequality for a large class of probability measures. Electron. Commun. Probab. 13 (2008) 60-66. | 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] F. Barthe, P. Cattiaux and C. Roberto. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and application to isoperimetry. Rev. Mat. Iberoam. 22 (2006) 993-1067. | MR | Zbl

[5] S. G. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999) 1903-1921. | MR | Zbl

[6] S. G. Bobkov and C. Houdré. Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997) 184-205. | MR | Zbl

[7] S. G. Bobkov and B. Zegarlinski. Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176 (2005) 69 pp. | MR | Zbl

[8] R. Carmona and A. Klein. Exponential moments for hitting times of uniformly ergodic Markov processes. Ann. Probab. 11 (1983) 648-655. | MR | Zbl

[9] P. Cattiaux. Calcul stochastique et opérateurs dégénérés du second ordre: I Résolvantes, théorème de Hörmander et applications. Bull. Sci. Math. 114 (1990) 421-462. | MR | Zbl

[10] P. Cattiaux. Calcul stochastique et opérateurs dégénérés du second ordre: II Problème de Dirichlet. Bull. Sci. Math. 115 (1991) 81-122. | MR | Zbl

[11] P. Cattiaux, P. Collet, A. Lambert, S. Martinez, S. Méléard and J. San Martin. Quasi-stationarity distributions and diffusion models in population dynamics. Ann. Probab. 37 (2009) 1926-1969. | MR | Zbl

[12] P. Cattiaux, N. Gozlan, A. Guillin and C. Roberto. Functional inequalities for heavy tailed distributions and application to isoperimetry. Electron. J. Probab. 15 (2010) 346-385. | MR | Zbl

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

[14] P. Cattiaux and A. Guillin. Trends to equilibrium in total variation distance. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 117-145. | Numdam | MR | Zbl

[15] P. Cattiaux and A. Guillin. Functional inequalities via Lyapunov conditions. In Proceedings of the Summer School on Optimal Transport (Grenoble 2009). To appear, 2010. Available at arXiv:1001.1822.

[16] P. Cattiaux, A. Guillin, F. Y. Wang and L. Wu. Lyapunov conditions for super Poincaré inequality. J. Funct. Anal. 256 (2009) 1821-1841. | MR | Zbl

[17] P. Cattiaux and S. Méléard. Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non extinction. J. Math. Biol. 60 (2010) 797-829. | MR | Zbl

[18] M.-F. Chen. Eigenvalues, Inequalities, and Ergodic Theory. Probab. Appl. (N. Y.). Springer, London, 2005. | MR | Zbl

[19] R. Douc, G. Fort and A. Guillin. Subgeometric rates of convergence of f-ergodic strong Markov processes. Stochastic Process. Appl. 119 (2009) 897-923. | 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] P. Fougères. Spectral gap for log-concave probability measures on the real line. In Séminaire de Probabilités XXXVIII 95-123. Lecture Notes in Math. 1857. Springer, Berlin, 2005. | MR | Zbl

[22] M. Hairer and J. C. Mattingly. Yet another look at Harris' ergodic theorem for Markov chains. Preprint, 2008. Available at arXiv:0810.2777. | MR | Zbl

[23] M. Ledoux. Spectral gap, logarithmic Sobolev constant, and geometric bounds. In Surveys in Differential Geometry, Vol. IX 219-240. Int. Press, Somerville, MA, 2004. | MR | Zbl

[24] D. Loukianova, O. Loukianov and S. Song. Poincaré inequality and exponential integrability of hitting times for linear diffusions. Available at ArXiv:0907.0762, 2009. | Numdam | Zbl

[25] P. Mathieu. Hitting times and spectral gap inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 33 (1997) 437-465. | Numdam | MR | Zbl

[26] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer, London, 1993. | MR | Zbl

[27] M. Röckner and F. Y. Wang. Weak Poincaré inequalities and L 2 -convergence rates of Markov semi-groups. J. Funct. Anal. 185 (2001) 564-603. | MR | Zbl

[28] F. Y. Wang. Functional Inequalities, Markov Processes and Spectral Theory. Science Press, Beijing, 2005.

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