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.
Mots clés : poincaré inequalities, Lyapunov functions, hitting times, log-concave measures, Poincaré-Sobolev inequalities
@article{AIHPB_2013__49_1_95_0, author = {Cattiaux, Patrick and Guillin, Arnaud and Zitt, Pierre Andr\'e}, title = {Poincar\'e inequalities and hitting times}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {95--118}, publisher = {Gauthier-Villars}, volume = {49}, number = {1}, year = {2013}, doi = {10.1214/11-AIHP447}, mrnumber = {3060149}, zbl = {1270.26018}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP447/} }
TY - JOUR AU - Cattiaux, Patrick AU - Guillin, Arnaud AU - Zitt, Pierre André TI - Poincaré inequalities and hitting times JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 95 EP - 118 VL - 49 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP447/ DO - 10.1214/11-AIHP447 LA - en ID - AIHPB_2013__49_1_95_0 ER -
%0 Journal Article %A Cattiaux, Patrick %A Guillin, Arnaud %A Zitt, Pierre André %T Poincaré inequalities and hitting times %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 95-118 %V 49 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP447/ %R 10.1214/11-AIHP447 %G en %F AIHPB_2013__49_1_95_0
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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