On the approximation for singularly perturbed stochastic wave equations

Abstract

We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations \nu u_{tt}+u_t=\D u+f(u)+\nu^\alpha\dot{W} on a bounded spatial domain. An asymptotic approximation to the stochastic wave equation is constructed by a special transformation and splitting of $\nu u_{t}$. This splitting gives a clear description of the structure of $u$. The approximating model, for small $\nu>0$\,, is a stochastic nonlinear heat equation for exponent $0\leq\alpha<1$\,, and is a deterministic nonlinear wave equation for exponent $\alpha>1$\,.