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We here show how the methods recently applied by [DW16] to solve thestochastic nonlinear Schr\"odinger equation on $\mathbb{T}^2$ can be enhancedto yield solutions on $\mathbb{R}^2$ if the non-linearity is weak enough. Weprove that the solutions remains localized on compact time intervals whichallows us to apply energy methods on the full space.

Solution to the stochastic Schr\"odinger equation on the full space