Singular Perturbation Approximation Problem Of Random Vibration Equation

In this paper, an averaging method is applied to derive effective approximation to the following singularly perturbed stochastic vibration equation εuttε(t)+utε(t)=f(uε(t))+εαW(t), uε(0)=u0∈Rn,utε(0)=ul∈Rn. n dimensional Q Wiener process defined on a complete probability space (Ω,F,{Ft}t≥0,P) and ε is a parameter characterizing the singular perturbation, εα parametrizes the strength of the noise. Averaging approximation method and martingale approximation method yield the following effective approximation when ut=f(u), u(0)＝uo∈Rn and the more effective approximation when0≤α≤1/2utε=f(uε)+εαW,uε(0)=u0∈Rn.