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

TL;DR

This paper extends methods for solving the stochastic nonlinear Schrödinger equation from a torus to the full space, demonstrating solution localization and energy estimates for weak non-linearity.

Contribution

It adapts existing techniques to the full space setting, enabling solutions on  with weak non-linearity and localized behavior.

Findings

Solutions remain localized on compact time intervals

Energy methods are applicable on  for weak non-linearity

Extension of solution techniques from  to 

Abstract

We here show how the methods recently applied by [DW16] to solve the stochastic nonlinear Schr\"odinger 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.