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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     title = {Poincar\'e inequalities and hitting times},
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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://www.numdam.org/articles/10.1214/11-AIHP447/

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