  We prove the global space-time bound for the mass critical nonlinear
Schr\"odinger equation perturbed by a small multiplicative noise in dimension
three. The associated scattering behavior are also obtained. We also prove a
global Strichartz space-time bound for the linear stochastic model, which is
new itself and serves a prototype model for the nonlinear case. The proof
combines techniques from \cite{fan2018global}, \cite{fan2020long} as well as
local smoothing estimates for linear Schr\"odinger operators.
