| We consider the operator differential equation perturbed by a fast Markov process: ddtue t=A&parl0;y&parl0; te&parr0;&parr0;ue t,t>0 ue0 =u0 in a separable Hilbert space H. Here y is an ergodic jump Markov process in phase space Y satisfying some mixing conditions and {A(y), y ∈ Y} is a family of closed linear operators. We study the asymptotic behavior of the distributions of uet/e . For the case when the operators A(y) commute, Salehi and Skorokhod (1996) proved that the distributions of uet/e asymptotically coincide with the distributions of some Gaussian random field with independent increments.;We do not assume that the operators A(y) commute, but we impose some conditions on the structure of these operators. We study the asymptotic behavior of the stochastic process zet=e -tA&d1;ue t , where a = ∫ A( y)rho(dy), and rho(·) is the ergodic distribution of the Markov process y(t), t ≥ 0. We prove that the stochastic process zet/e converges weakly as e→0 to a diffusion process z˜(t), t ≥ 0, which is described using its generator. The proof is based on the theorem on weak convergence of H-valued stochastic processes to a diffusion process. |
