Polynomial mixing for the 3D damped cubic nonlinear Schrödinger equation with degenerate noise
Abstract
We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schrödinger 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 H1 distance for every 1≤ p<∞. 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 L2 distance. A stationary regularity gain to H2− then enables us to upgrade the convergence to H1.
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BibTeX
@misc{liu2026polynomial,
title = {{Polynomial mixing for the 3D damped cubic nonlinear Schrödinger equation with degenerate noise}},
author = {Rongchang Liu and Kening Lu and Lin Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08645},
eprint = {2609.08645},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08645},
}
RIS
TY - UNPB
TI - Polynomial mixing for the 3D damped cubic nonlinear Schrödinger equation with degenerate noise
AU - Rongchang Liu
AU - Kening Lu
AU - Lin Shi
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08645
UR - https://arxiv.org/abs/2609.08645
AB - We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schrödinger 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≤ p<∞. 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.
ER -