An averaging principle for dynamical systems in Hilbert space with Markov random perturbations

We study the asymptotic behavior of solutions of differential equations dx[var epsilon](t)/dt = A(y(t/[var epsilon]))x[var epsilon](t), x[var epsilon](0) = x0, where A(y), for y in a space Y, is a family of operators forming the generators of semigroups of bounded linear operators in a Hilbert space H, and y(t) is an ergodic jump Markov process in Y. Let where [varrho](dy) is the ergodic distribution of y(t). We show that under appropriate conditions as [var epsilon] --> 0 the process x[var epsilon](t) converges uniformly in probability to the nonrandom function which is the solution of the equation and that converges weakly to a Gaussian random function for which a representation is obtained. Application to randomly perturbed partial differential equations with nonrandom initial and boundary conditions are included.