A perturbation theory or ergodic Markov chains and application to numerical approximations

Perturbations to Markov chains and Markov processes are considered. The unp erturbed problem is assumed to be geometrically ergodic in the sense usuall y established through the use of Foster Lyapunov drift conditions. The pert urbations are assumed to be uniform, in a weak sense, on bounded time inter vals. The long-time behavior of the perturbed chain is studied. Application s are given to numerical approximations of a randomly impulsed ODE, an Ito stochastic differential equation (SDE), and a parabolic stochastic partial differential equation (SPDE) subject to space-time Brownian noise. Existing perturbation theories for geometrically ergodic Markov chains are not read ily applicable to these situations since they require very stringent hypoth eses on the perturbations.