  We study the focusing stochastic nonlinear Schr\"odinger equation in one
spatial dimension with multiplicative noise, driven by a Wiener process white
in time and colored in space, in the $L^2$-critical and supercritical cases.
The mass ($L^2$-norm) is conserved due to the multiplicative noise defined via
the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first
investigate how the energy is affected by various spatially correlated random
perturbations. We then study the influence of the noise on the global dynamics
measuring the probability of blow-up versus scattering behavior depending on
various parameters of correlation kernels. Finally, we study the effect of the
spatially correlated noise on the blow-up behavior, and conclude that such
random perturbations do not influence the blow-up dynamics, except for shifting
of the blow-up center location. This is similar to what we observed in [32] for
a space-time white driving noise.
