Averaging approximation to singularly perturbed nonlinear stochastic wave equations

An averaging method is applied to derive effective approximation to the following singularly perturbed nonlinear stochastic damped wave equation u u_{tt}+u_t=D u+f(u)+ u^alphadot{W} on an open bounded domain $DsubsetR^n$,, $1leq nleq 3$,. Here $ u>0$ is a small parameter characterising the singular perturbation, and $ u^alpha$,, $0leq alphaleq 1/2$,, parametrises the strength of the noise. Some scaling transformations and the martingale representation theorem yield the following effective approximation for small $ u$, u_t=D u+f(u)+ u^alphadot{W} to an error of $ord{ u^alpha}$,.