Exponential and Uniform Ergodicity of Markov Processes

Abstract

General characterizations of geometric convergence for Markov chains in discrete time on a general state space have been developed recently in considerable detail. Here we develop a similar theory for $\varphi$-irreducible continuous time processes and consider the following types of criteria for geometric convergence: 1. the existence of exponentially bounded hitting times on one and then all suitably "small" sets; 2. the existence of "Foster-Lyapunov" or "drift" conditions for any one and then all skeleton and resolvent chains; 3. the existence of drift conditions on the extended generator $\tilde\mathscr{A}$ of the process. We use the identity $\tilde\mathscr{A}R_\beta = \beta(R_\beta - I)$ connecting the extended generator and the resolvent kernels $R_\beta$ to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of criteria 1-3. These conditions yield criteria for exponential convergence of unbounded as well as bounded functions of the chain. They enable us to identify the dependence of the convergence on the initial state of the chain and also to illustrate that in general some smoothing is required to ensure convergence of unbounded functions.