Hitting times, functional inequalities, lyapunov conditions and uniform ergodicity

TL;DR

This paper explores the deep connections between Lyapunov conditions, functional inequalities, and ergodic properties of Markov processes, establishing equivalences and characterizations that enhance understanding of their long-term behavior.

Contribution

It demonstrates that strong functional inequalities are equivalent to Lyapunov conditions and characterizes these conditions via exponential moments of hitting times, advancing the theoretical framework.

Findings

Strong functional inequalities are equivalent to Lyapunov conditions.

Lyapunov conditions are characterized by exponential moments of hitting times.

Unbounded Lyapunov conditions can imply uniform ergodicity.

Abstract

The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincar{\'e} inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by $\bullet$ showing that strong functional inequalities are equivalent to Lyapunov type conditions; $\bullet$ showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times. We also give some complement concerning the link between Lyapunov conditions and in-tegrability property of the invariant probability measure and as such transportation inequalities , and we show that some "unbounded Lyapunov conditions" can lead to uniform ergodicity, and coming down from infinity property.