EXPONENTIAL AND UNIFORM ERGODICITY OF MARKOV-PROCESSES

General characterizations of geometric convergence for Markov chains i n discrete time on a general state space have been developed recently in considerable detail. Here we develop a similar theory for phi-irred ucible continuous time processes and consider the following types of c riteria for geometric convergence: 1. the existence of exponentially b ounded hitting times on one and then all suitably ''small'' sets; 2. t he existence of ''Foster-Lyapunov'' or ''drift'' conditions for any on e and then all skeleton and resolvent chains; 3. the existence of drif t, conditions on the extended generator (A) over tilde of the process. We use the identity (A) over tilde R(beta) = beta(R(beta) - I) connec ting the extended generator and the resolvent kernels Rp to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of criteria 1-3. These conditions yield c riteria for exponential convergence of unbounded as well as bounded fu nctions 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 un bounded functions.