On the small-mass limit for stationary solutions of stochastic wave equations with state dependent friction

TL;DR

This paper proves that stationary solutions of certain stochastic damped wave equations with state-dependent friction converge to the invariant measure of a related parabolic equation as the mass approaches zero, extending previous short-term results to long-term behavior.

Contribution

It establishes the validity of the Smoluchowski-Kramers approximation for stationary solutions over long time scales in a class of stochastic wave equations.

Findings

Stationary solutions converge to the invariant measure of the limiting equation.

Convergence holds in Wasserstein distance with respect to the $H^{-1}$ norm.

The approximation is valid in the long time regime, not just fixed intervals.

Abstract

We investigate the convergence, in the small mass limit, of the stationary solutions of a class of stochastic damped wave equations, where the friction coefficient depends on the state and the noisy perturbation if of multiplicative type. We show that the Smoluchowski-Kramers approximation that has been previously shown to be true in any fixed time interval, is still valid in the long time regime. Namely we prove that the first marginals of any sequence of stationary solutions for the damped wave equation converge to the unique invariant measure of the limiting stochastic quasilinear parabolic equation. The convergence is proved with respect to the Wasserstein distance associated with the $H^{-1}$ norm.