POLYNOMIAL MIXING FOR A WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION

doi:10.1007/s10473-025-0513-z

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (5): 2029-2059.doi: 10.1007/s10473-025-0513-z

POLYNOMIAL MIXING FOR A WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION

Jing GUO, Zhenxin LIU^*

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

收稿日期:2023-09-14 修回日期:2024-10-31 出版日期:2025-09-25 发布日期:2025-10-14

POLYNOMIAL MIXING FOR A WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION

Jing GUO, Zhenxin LIU^*

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received:2023-09-14 Revised:2024-10-31 Online:2025-09-25 Published:2025-10-14
Contact: ^*Zhenxin Liu, E-mail: zxliu@dlut.edu.cn
About author:Jing Guo, E-mail: jingguo062@hotmail.com
Supported by:
Liu's research was supported by the National Key R&D Program of China (2023YFA1009200), the NSFC (11925102) and the Liaoning Revitalization Talents Program (XLYC2202042).

摘要/Abstract

摘要： This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schrödinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing and defocusing nonlinearities, with exponents of the nonlinearity $\sigma\in[0,2)$ and $\sigma\in[0,\infty)$, and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method. In the focusing case, our result generalizes the earlier results in [12], where $\sigma=1$.

关键词: stochastic damped nonlinear Schrödinger equation, uniqueness of invariant measure, polynomial mixing, coupling, Girsanov theorem

Abstract: This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schrödinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing and defocusing nonlinearities, with exponents of the nonlinearity $\sigma\in[0,2)$ and $\sigma\in[0,\infty)$, and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method. In the focusing case, our result generalizes the earlier results in [12], where $\sigma=1$.

Key words: stochastic damped nonlinear Schrödinger equation, uniqueness of invariant measure, polynomial mixing, coupling, Girsanov theorem

中图分类号:

35Q55

Jing GUO, Zhenxin LIU. POLYNOMIAL MIXING FOR A WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION[J]. 数学物理学报(英文版), 2025, 45(5): 2029-2059.

Jing GUO, Zhenxin LIU. POLYNOMIAL MIXING FOR A WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION[J]. Acta mathematica scientia,Series B, 2025, 45(5): 2029-2059.

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POLYNOMIAL MIXING FOR A WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION

POLYNOMIAL MIXING FOR A WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATION

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