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

Generally, the convergence rate in $$L^2$$ L 2 -exponential ergodicity $$\lambda $$ λ is an upper bound for the convergence rate $$\kappa $$ κ in uniform ergodicity for a Markov process, that is, $$\lambda \geqslant \kappa $$ λ ⩾ κ . In this paper, we prove that $$\kappa \geqslant \inf \left\{ \lambda ,1/M_H\right\} $$ κ ⩾ inf λ , 1 / M H , where $$M_H$$ M H is a uniform bound on the moment of the hitting time to a “compact” set H. In the case where $$M_H$$ M H can be made arbitrarily small for H large enough. we obtain that $$\lambda =\kappa $$ λ = κ . The general results are applied to Markov chains, diffusion processes and solutions to stochastic differential equations driven by symmetric stable processes.