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

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.