Well-posedness of the focusing stochastic nonlinear Schrödinger equation: L²-critical and supercritical cases

We study the focusing L² -critical and supercritical stochastic nonlinear Schrödinger equation subject to additive or multiplicative noise. We investigate global or long time behavior of solutions in H¹ , 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 H¹ -valued initial data with positive energy in both cases.