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

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.