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

摘要

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.