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We study the focusing stochastic nonlinear Schr\"odinger equation in onespatial dimension with multiplicative noise, driven by a Wiener process whitein 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 viathe Stratonovich integral, the energy (Hamiltonian) is not preserved. We firstinvestigate how the energy is affected by various spatially correlated randomperturbations. We then study the influence of the noise on the global dynamicsmeasuring the probability of blow-up versus scattering behavior depending onvarious parameters of correlation kernels. Finally, we study the effect of thespatially correlated noise on the blow-up behavior, and conclude that suchrandom perturbations do not influence the blow-up dynamics, except for shiftingof the blow-up center location. This is similar to what we observed in [32] fora space-time white driving noise.

Behavior of solutions to the 1D focusing stochastic nonlinear Schr\"odinger equation with spatially correlated noise