Convergence Rates in Uniform Ergodicity by Hitting Times and L2-Exponential Convergence Rates

Generally, the convergence rate in L 2 -exponential ergodicity λ is an upper bound for the convergence rate κ in uniform ergodicity for a Markov process, that is, λ ⩾ κ . In this paper, we prove that κ ⩾ inf λ , 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 λ = κ . The general results are applied to Markov chains, diffusion processes and solutions to stochastic differential equations driven by symmetric stable processes.