Abstract
Global Well-Posedness of Stochastic Hartree Equations with Multiplicative Noise in the Mass- and Energy-Critical Cases
We consider the defocusing stochastic Hartree equation with finite-dimensional linear multiplicative noise in the mass- and energy-critical cases in dimensions . The proof is based on a Doss–Sussmann-type transformation, which converts the stochastic equation into a random dispersive equation with lower-order coefficients. We establish local well-posedness and a blow-up alternative in for the mass-critical case and in for the energy-critical case, using random Strichartz estimates and Hartree-type nonlinear estimates. Global well-posedness is then obtained through stability arguments on random time intervals, treating the transformed equation as a perturbation of the corresponding deterministic critical Hartree equation.