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

Abstract

We study the asymptotic behavior of the distributions of the solution of the differential equation of the form du ϵ(t) dt =A y( t ϵ u ϵ(t) 1 u ϵ(0)=u 0 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 u ̂ ϵ(t) which is expressed in terms of the solution of the averaged equation d u(t) dt = A u(t), u(0)=u 0 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 u ϵ(t/ϵ) and u ̂ ϵ(t) asymptotically coincide.