We investigate the global well-posedness of the stochastic Hartree equation with additive noise in both the mass-critical and energy-critical cases. The investigation begins with the establishment of the local well-posedness result using the fixed-point method, which is based on deterministic and stochastic Strichartz inequalities in suitable functional spaces. Then, probabilistic a priori bounds on the mass and energy of the solutions are derived through stochastic analysis. These bounds enable the construction of global solutions via an iterative application of the perturbation lemma. The main ingredient is to establish the global solutions of the stochastic Hartree equation in the critical cases under stochastic forcing.
