The paper is devoted to studying the stochastic nonlinear wave (NLW) equationThe equation is supplemented with the Dirichlet boundary condition. Here f is a nonlinear term, h(x) is a function in H 1 0 (D) and η(t, x) is a non-degenerate white noise. We show that the Markov process associated with the flow ξu(t) = [u(t),u(t)] has a unique stationary measure µ, and the law of any solution converges to µ with exponential rate in the dual-Lipschitz norm.

We consider the damped nonlinear wave (NLW) equation driven by a spatially regular white noise. 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 seems to be new in the context of randomly forced PDE's. The proof is based on an extension of methods developed in [JNPS] and [JNPS14] in the case of kick forced dissipative PDE's with parabolic regularisation 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). P t (v, Γ)σ(dv), t ≥ 0, 1 Some estimates for the H s -norm of the solutions are given in Section 5.2.

The paper is devoted to the derivation of the large deviations principle for the family of stationary measures of the Markov process generated by the flow of the damped nonlinear wave equations with vanishing white noise. One of the main novelties here is that we do not assume that the deterministic equation possesses a unique equilibrium and we do not impose roughness on the noise. We introduce a new mathematical tool called the generalized stationary measure, which, informally speaking, is a stationary measure that is not necessarily σ‐additive. We show that any Markov operator admits such a measure and use this to develop the Freidlin‐Wentzell and Khasminskii approaches to the infinite‐dimensional setting. We also extend Sowers' method when establishing exponential tightness. Some ingredients of the proof rely on rather nonstandard techniques.© 2017 Wiley Periodicals, Inc.

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The paper is devoted to studying the asymptotics of the family (µ ε )ε>0 of stationary measures of the Markov process generated by the flow of equatioṅand ϑ is a spatially regular white noise. By using the large deviations techniques, we prove that the family (µ ε ) is exponentially tight in H 1−γ (D) for any γ > 0 and vanishes exponentially outside any neighborhood of the set O of ωlimit points of the deterministic equation. In particular, any of its weak limits is concentrated on the closureŌ. A key ingredient of the proof is a new formula that allows to recover the stationary measure µ of a Markov process with good mixing properties, knowing only some local information about µ. In the case of trivial limiting dynamics, our result implies that the family (µ ε ) obeys the large deviations principle.