On the approximation for singularly perturbed stochastic wave equations

TL;DR

This paper investigates the asymptotic behavior of singularly perturbed stochastic wave equations, deriving simplified models like stochastic heat or deterministic wave equations depending on the scaling parameter.

Contribution

It introduces a novel transformation and splitting method to approximate the complex stochastic wave equations with simpler models based on the perturbation parameter.

Findings

For small perturbation parameter, the model approximates a stochastic heat equation.

Depending on the scaling exponent, the approximation becomes a deterministic wave equation.

The method clarifies the structure of solutions to 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 $\alpha>1$\,.