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

Abstract

Generally, the convergence rate in \(L^2\)-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 \(\kappa \geqslant \inf \left\{ \lambda ,1/M_H\right\} \), 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 stochastic differential equations driven by symmetric stable processes.