Global well-posedness of the 4-d energy-critical stochastic nonlinear Schr\"{o}dinger equations with non-vanishing boundary condition

TL;DR

This paper proves the global well-posedness and unconditional uniqueness of solutions for the 4-dimensional energy-critical stochastic nonlinear Schrödinger equation with non-vanishing boundary conditions, advancing understanding of stochastic PDEs in critical regimes.

Contribution

It establishes the first global well-posedness and uniqueness results for this class of stochastic Schrödinger equations with non-vanishing boundary conditions.

Findings

Global well-posedness in the energy space

Unconditional uniqueness of solutions

Perturbative approach to stochastic critical NLS

Abstract

We consider the energy-critical stochastic cubic nonlinear Schr\"odinger equation on $\mathbb R^4$ with additive noise, and with the non-vanishing boundary conditions at spatial infinity. By viewing this equation as a perturbation to the energy-critical cubic nonlinear Schr\"odinger equation on $\mathbb R^4$, we prove global well-posedness in the energy space. Moreover, we establish unconditional uniqueness of solutions in the energy space.