Long Time Behavior of Stochastic NLS with a Small Multiplicative Noise

We prove the global space-time bound for the defocusing mass critical nonlinear Schrödinger equation on R³ 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 Schrödinger equation on R at L² regularity, 2018) and Fan and Zhao (On long time behavior for stochastic nonlinear Schrödinger 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.