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

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-Solé [Memoirs of the AMS, Vol 199, 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 $ε\in]0,1]$, 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 Hölder norm.

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