Global Well-Posedness of the 4-D Energy-Critical Stochastic Nonlinear Schrödinger Equations with Non-Vanishing Boundary Condition

Abstract

We consider the energy-critical stochastic cubic nonlinear Schrodinger 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 Schrodinger 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.