2D Torus defocusing NLSE with a potential and random initial data

We consider the non-linear cubic defocusing Schrödinger equation on the 2D Torus ($T^2$) with a potential $V(x)\in H^{2^{+}}(T^2)$. The initial data, $u_0$,is given by a centered gaussian variable with covariance $(-\mathrm{\Delta }+V{)}^{-1}$, and thus is almost surely in $H^{0^{-}}(T^2)$. We show that it is almost surely well-possed for all time in the weak sense, and that $u-e^{it(\mathrm{\Delta }-V)}{u}_0$ is in $H^s(T^2)$ for some $s>0$, and that the solution is the limit of the solutions to the truncated equation. This result is a generalization of the results first given by Bourgain in Invariant Measures for the {2D}-Defocusing Nonlinear {Schr{ö}dinger} Equation.