Mean-field branching SDEs: propagation of chaos, scaling limits and phase transitions

We study branching SDEs with law-dependent immigration and their mean-field particle approximations. Under a dissipativity condition and sufficiently weak interaction, a uniform propagation-of-chaos bound in time of order $N^{-1\mathrm{/}2}$ is established. On every fixed finite time horizon, the same order of propagation of chaos holds for arbitrary finite interaction strength. A two-stage scaling limit connects continuous-time discrete-state mean-field birth–death processes to interacting branching diffusions and then to the nonlinear equation. For a logistic mean-field diffusion we prove a sharp criterion for extinction/non-extinction, and further show that weak enough interaction strength is necessary for a uniform-in-time approximation.