Quantitative weak propagation of chaos for McKean--Vlasov branching diffusion processes

We study in this paper the weak propagation of chaos for McKean–Vlasov diffusions with branching, whose induced marginal measures are nonnegative finite measures but not necessary probability measures. The flow of marginal measures satisfies a non-linear Fokker–Planck equation, along which we provide a functional Itô’s formula. We then consider a functional of the terminal marginal measure of the branching process, whose conditional value is solution to a Kolmogorov backward master equation. By using Itô’s formula and based on the estimates of second-order linear and intrinsic functional derivatives of the value function, we finally derive a quantitative weak convergence rate for the empirical measures of the branching diffusion processes with finite population.