Long time behavior of stochastic NLS with a small multiplicative noise

TL;DR

This paper establishes global bounds and scattering behavior for the mass-critical nonlinear Schrödinger equation with small multiplicative noise in three dimensions, introducing new space-time bounds for the linear stochastic model.

Contribution

It provides the first global Strichartz space-time bounds for the linear stochastic Schrödinger model and extends these results to the nonlinear case with small noise.

Findings

Proved global space-time bounds for the nonlinear stochastic Schrödinger equation.

Established scattering behavior in the presence of small multiplicative noise.

Introduced a new prototype model with global bounds for the linear stochastic Schrödinger equation.

Abstract

We prove the global space-time bound for the mass critical nonlinear Schr\"odinger equation perturbed by a small multiplicative noise in dimension three. The associated scattering behavior are also obtained. We also prove a global Strichartz space-time bound for the linear stochastic model, which is new itself and serves a prototype model for the nonlinear case. The proof combines techniques from \cite{fan2018global}, \cite{fan2020long} as well as local smoothing estimates for linear Schr\"odinger operators.