Ergodic results for the stochastic nonlinear Schr\"odinger equation with large damping

TL;DR

This paper investigates the stochastic nonlinear Schrödinger equation with large damping and additive noise, proving the uniqueness of the invariant measure in two and three dimensions, which advances understanding of its long-term behavior.

Contribution

It establishes the uniqueness of the invariant measure for the stochastic nonlinear Schrödinger equation with large damping, a novel result in this context.

Findings

Uniqueness of invariant measure proven for large damping

Results applicable in 2D and 3D settings

Enhances understanding of long-term stochastic dynamics

Abstract

We study the nonlinear Schr\"odinger equation with linear damping, i.e. a zero order dissipation, and additive noise. Working in $R^d$ with d = 2 or d = 3, we prove the uniqueness of the invariant measure when the damping coefficient is sufficiently large.