Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare

TL;DR

This paper explores the connection between Lyapunov methods and Poincaré inequalities in analyzing ergodic properties of continuous Markov processes, introducing new inequalities and applying them to diffusion processes.

Contribution

It establishes a link between Lyapunov controls and Poincaré inequalities via Lyapunov-Poincaré inequalities, with explicit diffusion process examples and improvements over existing results.

Findings

New inequalities linking Lyapunov and Poincaré methods

Explicit examples for diffusion processes

Enhanced results for kinetic Fokker-Planck equations

Abstract

We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincar\'e type). We show that they can be linked through new inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier, Helffer-Nier and Villani is in particular discussed in the final section.