We prove the global space-time bound for the defocusing mass critical nonlinear Schr？dinger equation on \({\mathbb {R}}^3\) perturbed by a small multiplicative noise. The associated scattering behavior is also obtained. In addition to techniques from Fan and Xu (Global well-posedness for the defocusing mass-critical stochastic nonlinear Schr？dinger equation on \(\mathbb {R}\) at \({L}^{2} \) regularity, 2018) and Fan and Zhao (On long time behavior for stochastic nonlinear Schr？dinger equations with a multiplicative noise, 2020), the main new ingredients are the decomposition of the solution tailored for the bootstrap argument in this problem, and the incorporation of local smoothing norms to close the argument. We also prove the global space-time Strichartz estimate for the linear stochastic equation. It is a toy model of our nonlinear problem, but the bound itself is new and of its own interest. Furthermore, the proof we give for the linear model is more direct, and also illustrates the proof strategy for the nonlinear problem.  

　　Publication:  

　　Communications in Mathematical Physics, volume 404, pages563–595 (2023)  

　　http://dx.doi.org/10.1007/s00220-023-04848-w  

　　Author:  

　　Chenjie Fan  

　　Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China  

　　Email: fancj@amss.ac.cn  

　　Weijun Xu  

　　Beijing International Center for Mathematical Research, Peking University, Beijing, China  

　　Zehua Zhao  

　　Department of Mathematics and Statistics, Beijing Institute of Technology, Key Laboratory of Algebraic Lie Theory and Analysis of Ministy of Education, Beijing, China