Convergence Rates in Uniform Ergodicity by Hitting Times and $L^2$-exponential Convergence Rates

TL;DR

This paper establishes new bounds relating exponential and uniform ergodicity rates of Markov processes, linking convergence rates to hitting times and $L^2$-exponential convergence, with applications to various stochastic processes.

Contribution

It proves a lower bound for uniform ergodicity rate in terms of hitting times and exponential convergence, extending understanding of convergence behavior in Markov processes.

Findings

Established a lower bound for uniform ergodicity rate using hitting times.

Showed conditions under which exponential and uniform ergodicity rates are equal.

Applied results to Markov chains, diffusions, and SDEs driven by stable processes.

Abstract

Generally the convergence rate in exponential ergodicity $\lambda$ is an upper bound for the convergence rate $\kappa$ in uniform ergodicity for a Markov process, that is $\lambda\geqslant\kappa$. In this paper, we prove that $\kappa\geqslant \inf \{lambda,1/M_H\}$, where $M_H$ is a uniform bound on the moment of the hitting time to a "compact" set $H$. In the case where $M_H$ can be made arbitrarily small for $H$ large enough, we obtain that $\lambda=\kappa$. The general results are applied to Markov chains, diffusion processes and solutions to SDEs driven by symmetric stable processes.