Global well-posedness for the mass-critical stochastic nonlinear Schr\"odinger equation on $\mathbb{R}$: general $L^2$ data

TL;DR

This paper proves global well-posedness for the stochastic defocusing mass-critical nonlinear Schrödinger equation on the real line with arbitrary $L^2$ initial data, using stability results and extending to other stochastic NLS models.

Contribution

It establishes the first global well-posedness result for stochastic mass-critical NLS with general $L^2$ data, incorporating stability techniques and broadening understanding of stochastic NLS models.

Findings

Global well-posedness for stochastic NLS with $L^2$ data

Development of stability results for deterministic modified NLS

Extension to other stochastic NLS models

Abstract

We continue our study for the stochastic defocusing mass crtical nonlinear Schr\"odinger equation with conservative multiplicative noise, and show that it is globally well-posed for arbitrary initial data in $L_{\omega}^{\infty}L_{x}^{2}$. The main ingredients are several stability type results for deterministic (modified) NLS, which have their own interest. We also give some results on other stochastic NLS type models.