On asymptotic behavior of oscillatory solutions of operator differential equations perturbed by a fast Markov process

We study the asymptotic behavior of the distributions of the solution of the differential equation of the form duϵ(t)dt=Ay(tϵuϵ(t)1 uϵ(0)=u0 in a separable Hilbert space H where y(t) is an ergodic homogenous Markov process in a measurable space (Y, C) satisfying some mixing conditions and A(y), y ε Y is a family of commuting closed linear operators with the same dense domain. Using the spectral representation of the solution we construct an H-valued process ûϵ(t) which is expressed in terms of the solution of the averaged equation du(t)dt=Au(t), u(0)=u0 where A = ƒ A(y)ϱ(dy) and ϱ is the ergodic distribution of Y(t), and some Gaussian random fields with independent increments. We show that the distributions of uϵ(t/ϵ) and ûϵ(t) asymptotically coincide.