Exponential mixing for the randomly forced NLS equation

TL;DR

This paper proves exponential mixing of the invariant measure for a randomly forced nonlinear Schrödinger equation, highlighting the importance of asymptotic compactness and control in understanding ergodic behavior of such systems.

Contribution

It extends previous work by establishing exponential mixing for the NLS with localized damping and noise, advancing the understanding of its ergodic properties.

Findings

Proves exponential mixing for the invariant measure

Highlights the role of asymptotic compactness and control

Extends prior results on dispersive equations

Abstract

This paper investigates exponential mixing of the invariant measure for randomly forced nonlinear Schr\"{o}dinger equation, with damping and random noise localized in space. Our study emphasizes the crucial role of exponential asymptotic compactness and control properties in establishing the ergodic properties of random dynamical systems. This work extends the series [15, 45] on the statistical behavior of randomly forced dispersive equations.