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 $(-Δ+V)^{-1}$, and thus is almost surely in $H^{0^-}(T^2)$. We show that it is almost sur