Stochastic nonlinear Schroedinger equations in the defocusing mass and energy critical cases

We study the stochastic nonlinear Schroedinger equations with linear multiplicative noise, particularly in the defocusing mass-critical and energy-critical cases. For general initial data, we prove the global existence and uniqueness of solutions in both cases. When the quadratic variation of noise is globally bounded, we also obtain the rescaled scattering behavior of solutions in the spaces $L^2$, $H^1$ as well as the pseudo-conformal space. Moreover, the Stroock-Varadhan type theorem for the topological support of solutions are also obtained in the Strichartz and local smoothing spaces.