Well-posedness of the focusing stochastic nonlinear Schr\"odinger equation: $L^2$-critical and supercritical cases

Abstract

We study the focusing $L^2$-critical and supercritical stochastic nonlinear Schr\"odinger equation subject to additive or multiplicative noise. We investigate global or long time behavior of solutions in $H^1$, which would correspond to global well-posedness in the deterministic case, with either deterministic or random initial data, and establish quantitative information about the well-posedness time, its probability and bounds on the solution in both cases. We then give criteria for finite time blow-up with positive probability for an $H^1$-valued initial data with positive energy in both cases.