A Smoluchowski-Kramers approximation for an infinite dimensional system with state-dependent damping

TL;DR

This paper investigates the approximation of infinite-dimensional wave equations with state-dependent damping and noise, showing convergence to a stochastic parabolic equation with an additional noise-induced drift in the small mass limit.

Contribution

It establishes the validity of a Smoluchowski-Kramers approximation for wave equations with state-dependent damping and multiplicative noise, revealing the emergence of a noise-induced drift.

Findings

Solution converges to a stochastic quasilinear parabolic equation in the small mass limit.

A noise-induced extra drift term appears in the limiting equation.

The approximation holds under specific conditions on damping and noise.

Abstract

We study the validity of a Smoluchowski-Kramers approximation for a class of wave equations in a bounded domain of $\mathbb{R}^n$ subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass limit the solution converges to the solution of a stochastic quasilinear parabolic equation where a noise-induced extra drift is created.