The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

TL;DR

This paper investigates the long-term statistical behavior of stochastic damped wave equations in 2D and 3D, demonstrating exponential convergence to a unique invariant measure and analyzing the small mass limit convergence to a reaction-diffusion equation.

Contribution

It establishes uniform exponential ergodicity for the damped wave equations with polynomial nonlinearities and proves the convergence of invariant measures in the small mass limit.

Findings

Exponential attraction to a unique invariant measure under sufficient stochastic forcing.

Convergence of invariant measures in Wasserstein distance as mass tends to zero.

Extension of small mass limit results to equations with polynomial nonlinearities.

Abstract

We study the long time statistics of a class of semi--linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction--diffusion equation. This together with uniform geometric ergodcity implies the validity of the small mass limit for the solutions on the infinite time horizon $[0,\infty)$, thereby extending previously known results established for the damped wave equations under Lipschitz nonlinearities.