Local large deviations for randomly forced nonlinear wave equations with localized damping

TL;DR

This paper establishes a large deviation principle for locally damped nonlinear wave equations with bounded noise, overcoming compactness issues by leveraging asymptotic properties of wave dynamics and introducing a novel LDP approach for random systems.

Contribution

It introduces a new method for proving large deviations in nonlinear wave equations with localized damping and non-degenerate noise, addressing the lack of smoothing effects.

Findings

LDP established for empirical distributions of the system

Overcomes compactness issues via asymptotic exponential tightness

Develops a new approach for LDP in random dynamical systems

Abstract

We study the large deviation principle (LDP) for locally damped nonlinear wave equations perturbed by a bounded noise. When the noise is sufficiently non-degenerate, we establish the LDP for empirical distributions with lower bound of a local type. The primary challenge is the lack of compactness due to the absence of smoothing effect. This is overcome by exploiting the asymptotic compactness for the dynamics of waves, introducing the concept of asymptotic exponential tightness for random measures, and establishing a new LDP approach for random dynamical systems.