Abstract
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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