Subgeometric rates of convergence for a class of continuous-time Markov process

Abstract

Let (Φt)t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function Pt(x, ·) to π; specifically, we find conditions under which r(t)||Pt(x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.