A limit theorem for stochastic initial value problems with multiscales

In this paper we deal with stochastic initial value problems which have three different scales of variations represented by a small parameter. We obtain a limit theorem of KHASMINSKII type on an asymptotically unbounded interval in a framework based on separable Banach spaces and evolution operators. The expectations of random propagators are approximated by a backward propagator solving a final value problem as the small parameter vanishes. This result can be applied to diffusion effects caused by multiple wave scattering in random media.