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

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 non-linear with Lipschitz continuous coefficients. Under suitable conditions on the covariance measure, Dalang and Sanz-Solé (Memoirs of the AMS, 199, 931, 2009) have proved the existence of a random field solution with Hölder continuous sample paths, jointly in both arguments, time and space. By perturbing the driving noise with a multiplicative parameter ε ∈]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 (Dupuis and Ellis, A weak convergence approach to the theory of large deviations, Wiley, 1997), we prove that this family satisfies a Laplace principle in the Hölder norm. © 2011 Springer-Verlag Berlin Heidelberg.