On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation

Abstract

We investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear Schrödinger equation. We prove that, contrary to the deterministic case, any initial data gives birth to a solution which develops singularities. Moreover, the singularities appear immediately. We use a stochastic generalization of the variance identity and a control argument.