A Perturbation Theory for Ergodic Properties of Markov Chains

Perturbations to Markov chains and Markov processes are considered. The unperturbed problem is assumed to be geometrically er-godic in the sense usually established through use of Foster-Lyapunov drift conditions. The perturbations are assumed to be uniform, in a weak sense, on bounded time intervals. The long-time behaviour of the perturbed chain is studied. Applications are given to numerical approximations of a randomly impulsed ODE, an It^ o SDE and a parabolic SPDE subject to space-time Brownian noise. Existing perturbation theories for geometrically ergodic Markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations.