Solution to the stochastic Schrödinger equation on the full space

Abstract

We here show how the methods recently applied by [DW16] to solve the stochastic nonlinear Schrödinger equation on ${\mathbb{T}}^2$ can be enhanced to yield solutions on ${\mathbb{R}}^2$ if the non-linearity is weak enough. We prove that the solutions remains localized on compact time intervals which allows us to apply energy methods on the full space.