A Laplace principle for a stochastic wave equation in spatial dimension three

TL;DR

This paper establishes a Laplace principle for solutions to a stochastic wave equation in three dimensions driven by Gaussian noise, extending large deviation theory to this nonlinear, spatially covariant setting.

Contribution

It proves a Laplace principle for the family of solutions to a stochastic wave equation with Gaussian noise, using the weak convergence approach in a three-dimensional spatial context.

Findings

The family of solutions satisfies a Laplace principle in the Hölder norm.

The work extends large deviation principles to nonlinear stochastic wave equations in three dimensions.

Conditions on the covariance measure ensure the existence of solutions with Hölder continuous paths.

Abstract

We consider a stochastic wave equation in spatial dimension three, driven by a Gaussian noise, white in time and with a stationary spatial covariance. The free terms are nonlinear with Lipschitz continuous coefficients. Under suitable conditions on the covariance measure, Dalang and Sanz-Sol\'e [Memoirs of the AMS, Vol 199, 2009] have proved the existence of a random field solution with H\"older continuous sample paths, jointly in both arguments, time and space. By perturbing the driving noise with a multiplicative parameter $\epsilon\in]0,1]$, a family of probability laws corresponding to the respective solutions to the equation is obtained. Using the weak convergence approach to large deviations developed in [P. Dupuis, R. S. Ellis, 1997], we prove that this family satisfies a Laplace principle in the H\"older norm.