  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.
