Exponential mixing for nonlinear Schr\"odinger equations perturbed by bounded degenerate noise

TL;DR

This paper proves exponential convergence to a unique invariant measure for nonlinear Schrödinger equations with bounded noise acting on two modes, using a novel coupling method and stability analysis.

Contribution

It introduces a new criterion for exponential mixing in nonlinear Schrödinger equations with degenerate noise, combining stability, smoothing, and control techniques.

Findings

Proves exponential mixing for the perturbed nonlinear Schrödinger equation.

Develops a new criterion for exponential mixing based on asymptotic compactness.

Combines global stability, nonlinear smoothing, and geometric control methods.

Abstract

We prove the exponential convergence to a unique invariant measure for locally damped nonlinear Schr\"odinger equations, perturbed by bounded noise acting on only two Fourier modes. To tackle the lack of smoothing effect, we introduce asymptotic compactness of linearized system to enhance the coupling method. Inspired by [14,33,39], we establish a new criterion for exponential mixing. Elements from global stability, nonlinear smoothing, and geometric control are combined when applying this criterion.