  This paper is devoted to the well-posedness of stochastic nonlinear
Schr\"odinger equations in the energy space H1(Rd), which is a natural
continuation of our recent work [1]. We consider both focusing and defocusing
nonlinearities and prove global well-posedness in H1(Rd), including also the
pathwise continuous dependence on initial conditions, with exponents exactly
the same as in the deterministic case. In particular, this work improves
earlier results in [4]. Moreover, the local existence, uniqueness and blowup
alternative are also established for the energy-critical case. The approach
presented here is mainly based on the rescaling approach already used in [1] to
study the L2 case and also on the Strichartz estimates established in [12] for
large perturbations of the Laplacian.
