Averaging approximation to singularly perturbed nonlinear stochastic wave equations

TL;DR

This paper applies an averaging method to derive an effective approximation for a singularly perturbed nonlinear stochastic wave equation, capturing the behavior as the perturbation parameter approaches zero.

Contribution

The paper introduces a novel averaging approach to approximate singularly perturbed stochastic wave equations with quantifiable error bounds.

Findings

Effective approximation with error of order ( u^) for small ( u)

Reduction of complex wave equations to simpler stochastic PDEs

Validation of approximation through scaling and martingale techniques

Abstract

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