On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior

TL;DR

This paper investigates the validity and stability of the Smoluchowski-Kramers approximation for semi-linear stochastic wave equations, analyzing its behavior over fixed and infinite time intervals, including effects on invariant measures and large deviations.

Contribution

It provides new insights into the small mass limit for stochastic wave equations, extending the approximation's validity to infinite time and linking it with large deviations and long-term behavior.

Findings

Approximation holds on fixed time intervals.

Stability of approximation over infinite time.

Connections with invariant measures and large deviations.

Abstract

We discuss here the validity of the small mass limit (the so-called Smoluchowski-Kramers approximation) on a fixed time interval for a class of semi-linear stochastic wave equations, both in the case of the presence of a constant friction term and in the case of the presence of a constant magnetic field. We also consider the small mass limit in an infinite time interval and we see how the approximation is stable in terms of the invariant measure and of the large deviation estimates and the exit problem from a bounded domain of the space of square integrable functions.