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 du(epsilon)(t)/dt = A(y(t/eps ilon))u(epsilon)(t), u(epsilon)(0) = u(0) in a separable Hilbert space H where y(t) is an ergodic homogeneous Markov process in a measurable space (Y,C) satisfying some mixing conditions and {A(y), y is an elem ent of Y} is a family of commuting closed linear operators with the sa me dense domain. Using the spectral representation of the solution we construct an H-valued process (u) over cap(epsilon)(t) which is expres sed in terms of the solution of the averaged equation d (u) over bar(t )/dt = (A) over bar (u) over bar(t), (u) over bar(0) = u(0) where (A) over bar = integral A(y)rho(dy) and rho is the ergodic distribution of Y(t), and some Gaussian random fields with independent increments. We show that the distributions of u(epsilon)(t/epsilon) and (u) over cap (epsilon)(t) asymptotically coincide. Copyright (C) 1996 Elsevier Scie nce Ltd.