Large deviations for a spatial average of stochastic heat and wave equations

TL;DR

This paper establishes large deviation principles for the spatial averages of solutions to one-dimensional stochastic heat and wave equations driven by Gaussian noise, using Malliavin calculus to analyze their covariance structures.

Contribution

It provides the first large deviation results for spatial averages of these stochastic PDEs, extending the understanding of their probabilistic behavior as the spatial domain grows.

Findings

Large deviation principles are proved for spatial averages as domain size increases.

Results include large deviation principles in the space of continuous functions.

The methods involve Malliavin calculus to handle nonlinear functionals of the solutions.

Abstract

We consider the one-dimensional stochastic heat and wave equations driven by Gaussian noises with constant initial conditions. We study the spatial average of the solutions on an interval of length $R$ and show that the family of laws of the spatial average satisfies the large deviation principle as $R$ goes to infinity. We also present the large deviation principle in the space of continuous functions. We prove these results using the tools of Malliavin calculus to evaluate the covariance of nonlinear functionals of the solution.