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.

In this paper, the aim is to identify uncertain topological structures of stochastic multi-weighted complex networks (SMWCN), in which both time-varying delayed coupling and non-delayed coupling are considered. Distinguished from the previous work, graph theory and Lyapunov method are combined to research topology identification of SMWCN. Combining LaSalle-type invariance principle for stochastic differential equations and some strategies of inequalities, theoretical results about topology identification and partial topology identification of SMWCN are obtained under adaptive control and adaptive pinning control, respectively. Furthermore, by using the drive-response concept, response network can reach synchronization with drive network. In the end, two numerical simulations are provided to demonstrate the effectiveness and correctness of theoretical results, in which typical L\”{u} chaotic system is used to describe the vertex’s dynamical behavior.