On the convergence of stationary solutions in the Smoluchowski-Kramers approximation of infinite dimensional systems

TL;DR

This paper proves that, in the small mass limit, the invariant measures of stochastic damped wave equations converge to those of the corresponding stochastic parabolic equations, establishing a rigorous connection between the two via the Smoluchowski-Kramers approximation.

Contribution

It demonstrates the convergence of invariant measures for a class of stochastic wave equations to those of the limiting parabolic equations, including new bounds uniform in mass.

Findings

Invariant measures converge in Wasserstein metric

Convergence holds for Lipschitz and polynomial nonlinearities

New uniform bounds for solutions of stochastic wave equations

Abstract

We prove the convergence, in the small mass limit, of statistically invariant states for a class of semi-linear damped wave equations, perturbed by an additive Gaussian noise, both with Lipschitz-continuous and with polynomial non-linearities. In particular, we prove that the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable Wasserstein metric to the unique invariant measure of the limiting stochastic semi-linear parabolic equation obtained in the Smoluchowski-Kramers approximation. The Wasserstein metric is associated to a suitable distance on the space of square integrable functions, that is chosen in such a way that the dynamics of the limiting stochastic parabolic equation is contractive with respect to such a Wasserstein metric. This implies that the limiting result is a consequence of the validity of a generalized Smoluchowski-Kramers limit at fixed times. The proof of such a generalized limit requires new delicate bounds for the solutions of the stochastic wave equation, that must be uniform with respect to the size of the mass.