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We consider the damped nonlinear wave (NLW) equation driven by a spatiallyregular white noise. Assuming that the noise is non-degenerate in all Fouriermodes, we establish a large deviations principle (LDP) for the occupationmeasures of the trajectories. The lower bound in the LDP is of a local type,which is related to the weakly dissipative nature of the equation and seems tobe new in the context of randomly forced PDE's. The proof is based on anextension of methods developed in \cite{JNPS-2012} and \cite{JNPS-2014} in thecase of kick forced dissipative PDE's with parabolic regularisation propertysuch as, for example, the Navier-Stokes system and the complex Ginzburg-Landauequations. We also show that a high concentration towards the stationarymeasure is impossible, by proving that the rate function that governs the LDPcannot have the trivial form (i.e., vanish on the stationary measure and beinfinite elsewhere).

Local large deviations principle for occupation measures of the damped nonlinear wave equation perturbed by a white noise