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

We study the asymptotic behavior of solutions of differential equations dx(epsilon)(t)/dt = A(y(t/epsilon))x(epsilon)(t), x(epsilon)(0)) = x(0), 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 (A) over bar = integral A(y)rho(dy) where rho(dy) is the ergodic distribution of y(t). We show that under appropriate conditions as epsilon-->0 the process x(epsilon)(t) converges uniformly in probability to the nonrandom function (x) over bar(t) which is the solution of the equation d (x) over bar(t)/dt = (A) over bar (x) over bar(t), (x) over bar(0) = x(0) and that epsilon(-1/2)(x(epsilon)(t) - (x) over bar(t)) converges weakly to a Gaussian random function (x) over tilde(t) for which a representation is obtained. Application to randomly perturbed partial differential equations with nonrandom initial and boundary conditions are included.