Polynomial mixing for the stochastic Schr\"odinger equation with large damping in the whole space

TL;DR

This paper investigates the long-term behavior of the stochastic nonlinear Schrödinger equation with large damping in three-dimensional space, demonstrating polynomial convergence rates to equilibrium using coupling and Strichartz estimates.

Contribution

It establishes polynomial mixing rates for the stochastic Schrödinger equation under large damping, a result previously unknown.

Findings

Solutions converge to equilibrium at polynomial rates

The approach uses coupling with pathwise Strichartz estimates

Unique ergodicity is confirmed in the large damping regime

Abstract

We study the long-time mixing behavior of the stochastic nonlinear Schr\"odinger equation in $\mathbb{R}^d$, $d\le 3$. It is well known that, under a sufficiently strong damping force, the system admits unique ergodicity, although the rate of convergence toward equilibrium has remained unknown. In this work, we address the mixing property in the regime of large damping and establish that solutions are attracted toward the unique invariant probability measure at polynomial rates of arbitrary order. Our approach is based on a coupling strategy with pathwise Strichartz estimates.