Polynomial mixing for the 3D damped cubic nonlinear Schr\"odinger equation with degenerate noise

Abstract

We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schr\"odinger equation on the three-dimensional torus under saturating smooth finite rank Brownian forcing. The mixing rate is measured in the $p$-Wasserstein metric induced by the $H^1$ distance for every $1\le p<\infty$. We also obtain sharp geometric characterizations of saturation. The proof is based on a polynomial mixing criterion built on a stable--compact decomposition of the exact solution differences with polynomial moment control of the logarithmic path amplification. Dense Malliavin range allows the compact defect to be compensated by finite-dimensional Cameron--Martin shifts, producing a block multiplier with negative mean logarithm. A logarithmic transportation gauge, combined with a renewal--reset coupling scheme, then yields mixing at every prescribed polynomial order in the weaker $L^2$ distance. A stationary regularity gain to $H^{2-}$ then enables us to upgrade the convergence to $H^1$.