Global well-posedness for the mass-critical stochastic nonlinear Schr\"{o}dinger equation on $\mathbb{R}$: small initial data

TL;DR

This paper establishes the global existence and stability of solutions for the small data mass-critical stochastic nonlinear Schrödinger equation in one dimension, advancing understanding of stochastic PDEs at criticality.

Contribution

It introduces a novel approach to prove global well-posedness for the critical stochastic NLS with small initial data, including a truncation and limiting procedure.

Findings

Proved global existence of solutions for small initial data.

Demonstrated stability of solutions under initial data perturbations.

Established uniform bounds enabling criticality analysis.

Abstract

We prove the global existence of solution to the small data mass critical stochastic nonlinear Schr\"{o}dinger equation in $d=1$. We further show the stability of the solution under perturbation of initial data. Our construction starts with the existence of the solution to the truncated subcritical problem. We then obtain uniform bounds on these solutions that enable us to reach criticality and then remove the truncation.