Local large deviations principle for occupation measures of the stochastic damped nonlinear wave equation.

*(English. French summary)*Zbl 06996557Summary: We consider the damped nonlinear wave (NLW) equation driven by a noise which is white in time and colored in space. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures 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 is a novelty in the context of randomly forced PDE’s. The proof is based on an extension of methods developed in (Comm. Pure Appl. Math. 68 (12) (2015) 2108–2143) and (Large deviations and mixing for dissipative PDE’s with unbounded random kicks (2014) Preprint) in the case of kick forced dissipative PDE’s with parabolic regularization property such as, for example, the Navier-Stokes system and the complex Ginzburg-Landau equations. We also show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere).

##### MSC:

35R60 | PDEs with randomness, stochastic partial differential equations |

35L70 | Second-order nonlinear hyperbolic equations |

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |

60F10 | Large deviations |

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\textit{D. Martirosyan} and \textit{V. Nersesyan}, Ann. Inst. Henri Poincaré, Probab. Stat. 54, No. 4, 2002--2041 (2018; Zbl 06996557)

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