Solution to the stochastic Schrödinger equation on the full space

Abstract

We show here how the methods recently applied in Debussche and Weber (2018 Electron. J. Probab. 23 28) to solve the stochastic nonlinear Schrödinger equation on can be enhanced to yield solutions on if the nonlinearity is weak enough. We prove that the solutions remain localized on compact time intervals which allow us to apply energy methods on the full space.