We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrödinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability.
"Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise." Ann. Probab. 33 (3) 1078 - 1110, May 2005. https://doi.org/10.1214/009117904000000964