Exponential mixing for randomly forced wave and dispersive equations

Abstract

In a series of recent works, we explore the statistical properties of dispersive equations. This project highlights temphasizes the crucial role of dispersive nonlinear smoothing and control properties. We begin by establishing a general criterion for exponential mixing of random dynamical systems. This criterion is then applied to NLW, NLS, KdV, with damping and random noise localized in space.  The verification of this criterion is naturally linked to topics in deterministic PDEs, such as exponential asymptotic compactness in dynamical systems, global stability of the locally damped equations, and the control properties.