Approximating dynamics of a singularly perturbed stochastic wave equation with a random dynamical boundary condition

TL;DR

This paper investigates how a singularly perturbed stochastic wave equation with a random boundary condition can be approximated by simpler equations, depending on the perturbation parameter, using a splitting method to analyze their probabilistic behavior.

Contribution

It introduces a novel approximation method for a complex stochastic wave equation with boundary randomness, revealing different limiting equations based on the perturbation scale.

Findings

Approximate equations are stochastic parabolic or deterministic hyperbolic depending on the perturbation exponent.

The splitting method effectively derives the limiting behavior of the system.

The results provide insight into the dynamics of stochastic wave equations with boundary conditions.

Abstract

This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting skill is used to derive the approximating equation of the system in the sense of probability distribution, when the singular perturbation parameter is sufficiently small. The approximating equation is a stochastic parabolic equation when the power exponent of singular perturbation parameter is in $[1/2, 1)$, but a deterministic hyperbolic (wave) equation when the power exponent is in $(1, +\infty)$.