On the stochastic nonlinear Schrödinger equations at critical regularities

Abstract

We consider the Cauchy problem for the defocusing stochastic nonlinear Schrödinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by the first author with Bényi and Pocovnicu (2015) to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.

23 pages. Appendix A added. Published in Stoch. Partial Differ. Equ. Anal. Comput