带退化噪声的三维阻尼三次非线性薛定谔方程的多项式混合
Polynomial mixing for the 3D damped cubic nonlinear Schrödinger equation with degenerate noise
- Sichuan University（四川大学）
- University of Electronic Science and Technology of China（电子科技大学）
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AI总结：
该研究证明了三维环面上带饱和有限秩布朗强迫的散焦阻尼三次随机非线性薛定谔方程在$p$-Wasserstein度量下具有多项式混合性，通过稳定-紧致分解和耦合方案实现，并给出饱和的几何刻画。
AI中文摘要：
我们证明了在三维环面上，在饱和光滑有限秩布朗强迫下，散焦阻尼三次随机非线性薛定谔方程具有多项式混合性。混合速率在由$H^1$距离诱导的$p$-Wasserstein度量下测量，对所有$1\le p<\infty$成立。我们还获得了饱和的尖锐几何刻画。证明基于一个多项式混合准则，该准则建立在精确解差的稳定-紧致分解之上，并对对数路径放大进行多项式矩控制。稠密的Malliavin范围允许紧致缺陷通过有限维Cameron-Martin平移来补偿，产生具有负平均对数的块乘子。一个对数输运规范，结合更新-重置耦合方案，在较弱的$L^2$距离下以每个指定的多项式阶产生混合。平稳正则性提升到$H^{2-}$使我们能够将收敛升级到$H^1$。
英文摘要：
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 $1\le p<\infty$. 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 $H^1$.
