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

We study the asymptotic behavior of solutions of differential equations dxε(t)dt = A(y(tε))xε(t), xε(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 Ā = ∫ A(y)ϱ(dy) where ϱ(dy) is the ergodic distribution of y(t). We show that under appropriate conditions as ε → 0 the process xε(t) converges uniformly in probability to the nonrandom function x̄(t) which is the solution of the equation dx̄(t)dt = Āx̄(t), x̄(0) = x0 and that ε−12(xε(t) - x̄(t)) converges weakly to a Gaussian random function x̃(t) for which a representation is obtained. Application to randomly perturbed partial differential equations with nonrandom initial and boundary conditions are included.