Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Poincaré inequalities and hitting times

Patrick Cattiaux, Arnaud Guillin, and Pierre André Zitt

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 95-118.

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470127

Digital Object Identifier
doi:10.1214/11-AIHP447

Mathematical Reviews number (MathSciNet)
MR3060149

Zentralblatt MATH identifier
1270.26018

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60G10: Stationary processes 60J60: Diffusion processes [See also 58J65]

Keywords
Poincaré inequalities Lyapunov functions Hitting times Log-concave measures Poincaré–Sobolev inequalities

Citation

Cattiaux, Patrick; Guillin, Arnaud; Zitt, Pierre André. Poincaré inequalities and hitting times. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 95--118. doi:10.1214/11-AIHP447. https://projecteuclid.org/euclid.aihp/1359470127


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