Global energy solutions to a stochastic Schrödinger-Poisson system with multiplicative noise in two dimensions

We are concerned with the Cauchy problem for a two-dimensional stochastic Schrödinger-Poisson system with a multiplicative noise in the energy space. Depending on the values of the physical parameters, this system models a self-gravitating Bose-Einstein condensate of the dark matter in the universe or the electron transport in semiconductors under the effect of thermal fluctuations. We show that in both cases the system admits a unique global solution in H1. The key ingredient of this paper is to establish appropriate commutator estimates for the multiplicative noise.