We explore the statistical properties of dispersive equations. This study highlights the role of control properties and nonlinear smoothing in deterministic models to the ergodicity of random dynamical systems.
We begin by establishing a new criterion for exponential mixing and large deviations of random dynamical systems. This criterion is then applied to randomly forced nonlinear wave equations and nonlinear Schrödinger equations with degenerate damping, critical nonlinearity, and physically localized noise. The verification of this criterion is naturally linked to topics in deterministic systems, such as exponential asymptotic compactness in dynamical systems, global stability of the locally damped equations, and the controllability and stabilization properties.
