A Smoluchowski-Kramers approximation for the stochastic variational wave equation

TL;DR

This paper studies how solutions of a stochastic variational wave equation with damping and noise can be approximated by simpler parabolic equations, providing insights into their convergence behavior.

Contribution

It establishes the convergence of dissipative solutions of a stochastic wave equation to solutions of a stochastic parabolic equation under the Smoluchowski-Kramers approximation.

Findings

Weak solutions converge to stochastic parabolic solutions

Dissipative solutions exhibit stability under approximation

Provides a rigorous mathematical framework for the approximation process

Abstract

We investigate the Smoluchowski-Kramers approximation for the one-dimensional periodic variational wave equation with state-dependent damping and additive noise. We show that weak ``dissipative'' solutions converge to solutions of a stochastic quasilinear parabolic equation.