Polynomial mixing for the stochastic Schrödinger equation with large damping in the whole space

TL;DR

The paper addresses the mixing behavior of the stochastic nonlinear Schrödinger equation on $\mathbb{R}^d$ with $d\le 3$ in the regime of large damping. It develops a coupling approach enhanced by pathwise Strichartz estimates and Lyapunov-type controls to prove the existence of a unique invariant measure and to establish an algebraic (polynomial) rate of convergence to equilibrium for a broad class of observables. The main contributions include extending unique ergodicity and providing polynomial mixing rates in the unbounded domain, with refined nonlinear and noise-regularity assumptions (notably $Q:U\to H^3$ for mixing and $0<\sigma<\tfrac{3}{2}$ in $d=3$). This work bridges results between bounded domains and the whole space for dispersive SPDEs and highlights how large damping enables effective coupling without invoking Girsanov transforms. The methods and estimates presented have potential implications for understanding long-time behavior of dispersive SPDEs under strong dissipation in unbounded settings.

Abstract

We study the long-time mixing behavior of the stochastic nonlinear Schrödinger 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.

Theorems & Definitions (42)

• Theorem 1.1
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Remark 2.6
• Theorem 2.8
• Remark 2.9
• Lemma 3.1
• proof
• Lemma 3.2