Mixing and large deviations for nonlinear wave equation with white noise

TL;DR

This thesis investigates ergodicity, large deviations, and asymptotic behaviors of the stochastic nonlinear wave equation with white noise in 3D, establishing unique stationary measures, convergence properties, and large deviations principles.

Contribution

It proves the existence of a unique stationary measure for the stochastic NLW, establishes large deviations principles for stationary measures, and analyzes large time asymptotics of occupation measures.

Findings

Unique stationary measure attracts solutions exponentially.

Stationary measures satisfy large deviations principles.

Occupation measures exhibit local large deviations, with high concentration being impossible.

Abstract

This thesis is devoted to the study of ergodicity and large deviations for the stochastic nonlinear wave (NLW) equation with smooth white noise in 3D. Under some standard growth and dissipativity assumptions on the nonlinearity, we show that the Markov process associated with the flow of NLW equation has a unique stationary measure that attracts the law of any solution with exponential rate. This result implies, in particular, the strong law of large numbers as well as the central limit theorem for the trajectories. We next consider the problem of small noise asymptotics for the family of stationary measures and prove that this family obeys the large deviations principle. When the limiting equation (i.e., without noise) possesses finitely many stationary solutions, among which only one asymptotically stable solution "u", this result implies that the family of these measures weakly converges to the Dirac measure concentrated at "u". Finally, we study the problem of large time asymptotics for the family of occupation measures corresponding to the NLW equation and show that it satisfies the local large deviations principle. We also show that a high concentration towards the stationary measure is impossible by proving that the corresponding rate function does not have the trivial form.