Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space.

ABSTRACT

We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43)11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4)1147-1174, 2019) with a sub-quadratic nonlinearity.