Poincaré inequalities and hitting times

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. © 2013 Association des Publications de l'Institut Henri Poincaré.