Global well-posedness of the energy-critical stochastic nonlinear Schr\"odinger equation on the three-dimensional torus

TL;DR

This paper proves the global well-posedness of the energy-critical stochastic nonlinear Schrödinger equation on a three-dimensional torus with additive noise, using advanced probabilistic and functional analysis techniques.

Contribution

It is the first to establish global well-posedness for the periodic energy-critical SNLS in the energy space, adapting atomic spaces and probabilistic perturbation methods.

Findings

Established global well-posedness of the energy-critical SNLS on the 3D torus.

Introduced novel adaptation of atomic spaces for stochastic PDEs.

First result of its kind for periodic critical SNLS.

Abstract

We study the Cauchy problem of the defocusing energy-critical stochastic nonlinear Schr\"odinger equation (SNLS) on the three dimensional torus, forced by an additive noise. We adapt the atomic spaces framework in the context of the energy-critical nonlinear Schr\"odinger equation, and employ probabilistic perturbation arguments in the context of stochastic PDEs, establishing the global well-posedness of the defocusing energy-critical quintic SNLS in the energy space. It is the first global well-posedness result for the periodic SNLS in a critical space.