Local well-posedness holds for the energy-critical stochastic NLS in appropriate Sobolev spaces; in the focusing case with small noise intensity, positive-energy solutions blow up before a fixed positive time with positive probability.
Cazenave,Semilinear Schr¨ odinger equations, Courant Lecture Notes in Mathematics, 10
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The energy-critical stochastic nonlinear Schr\"odinger equation: well-posedness and blow-up
Local well-posedness holds for the energy-critical stochastic NLS in appropriate Sobolev spaces; in the focusing case with small noise intensity, positive-energy solutions blow up before a fixed positive time with positive probability.