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We consider the Cauchy problem for the defocusing stochastic nonlinearSchr\"odinger equations (SNLS) with an additive noise in the mass-critical andenergy-critical settings. By adapting the probabilistic perturbation argumentemployed in the context of the random data Cauchy theory by the first authorwith B\'enyi and Pocovnicu (2015) to the current stochastic PDE setting, wepresent a concise argument to establish global well-posedness of themass-critical and energy-critical SNLS.

On the stochastic nonlinear Schr\"odinger equations at critical regularities