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 It\^o'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 It\^o'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.