Poincar\'e inequalities and hitting times

TL;DR

This paper explores the relationships between Poincaré inequalities, hitting times, and Lyapunov conditions for reversible diffusion processes, extending known results and analyzing various functional inequalities under different hitting time integrability conditions.

Contribution

It establishes quantitative correspondences between spectral gap, hitting times, and Lyapunov conditions for reversible diffusions, and generalizes Poincaré constant results to superlinear potentials.

Findings

Equivalence between spectral gap, hitting times, and Lyapunov conditions with quantitative bounds.

Extension of Poincaré constant bounds to superlinear potentials for log-concave measures.

Ultracontractivity characterized by bounded Lyapunov conditions in one-dimensional cases.

Abstract

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions are well known. We give here the correspondance (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 Poincar\'e constant for logconcave 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.