Wellposedness of the cubic Gross-Pitaevskii equation with spatial white noise on R^2

In this paper, we prove the global wellposedness of the Gross-Pitaevskii equation with white noise potential, i.e. a cubic nonlinear Schrödinger equation with harmonic confining potential and spatial white noise multiplicative term. This problem is ill-defined and a Wick renormalization is needed in order to give a meaning to solutions. In order to do this, we introduce a change of variables which transforms the original equation into one with less irregular terms. We construct a solution as a limit of solutions of the same equation but with a regularized noise. This convergence is shown by interpolating between a diverging bound in a high regularity Hermite-Sobolev space and a Cauchy estimate in L^2(R^2)