发表机构
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学-华东师范大学上海数学与交叉学科研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究实直线上弱阻尼随机非线性薛定谔方程，在噪声非退化条件下证明唯一平稳测度存在及多项式混合，方法结合耦合论证、Foiaş-Prodi 估计与截断 Poincaré 不等式。
AI 中文摘要
我们考虑实直线上的弱阻尼随机非线性薛定谔（NLS）方程，该方程由时间上为白噪声、空间上光滑的噪声驱动。假设噪声充分非退化，我们证明该方程在集中于 $H^2$ 上的概率测度类中存在唯一的平稳测度，并在 $H^1$ 上的对偶-Lipschitz 度量下建立多项式混合。我们不对方程的阻尼大小施加任何限制。证明基于耦合论证，其关键要素是在 $H^1$ 范数下的 Foiaş-Prodi 型估计，该估计通过一个适应于线性化 NLS 动力学的 Lyapunov 泛函推导得出。为弥补紧致性的缺失，我们将该估计与截断的 Poincaré 不等式以及一个量化解的空间衰减的空间-时间权重函数相结合。
英文摘要
We consider the weakly damped stochastic nonlinear Schrödinger (NLS) equation on the real line, driven by a noise that is white in time and smooth in space. Assuming that the noise is sufficiently non-degenerate, we prove that the equation has a unique stationary measure in the class of probability measures concentrated on $H^2$, and establish polynomial mixing in the dual-Lipschitz metric over $H^1$. We do not impose any restriction on the size of the damping. The proof is based on a coupling argument, whose key ingredient is a Foiaş-Prodi-type estimate in the $H^1$-norm, derived by means of a Lyapunov functional adapted to the linearized NLS dynamics. To compensate for the loss of compactness, we combine this estimate with a truncated Poincaré inequality and a space-time weight function quantifying the spatial decay of solutions.
