Nonlinear Schrödinger equation with a white-noise potential: Phase-space approach to spread and singularity

We have also found that in the critical and supercritical focusing cases the presence of a white-noise random potential results in different conditions for singularity-with-positive-probability from the homogeneous case but does not prevent singularity formation. We show that in the supercritical focusing case the ensemble-averaged self-interaction energy and the momentum variance can exceed any fixed level in a finite time with positive probability.