Weak convergence analysis in the particle limit of the McKean--Vlasov equations using stochastic flows of particle systems

We present a proof showing that the weak error of a system of $n$ interacting stochastic particles approximating the solution of the McKean-Vlasov equation is $\mathcal{O}(n^{-1})$. Our proof is based on the Kolmogorov backward equation for the particle system and bounds on the derivatives of its solution, which we derive more generally using the variations of the stochastic particle system. The convergence rate is verified by numerical experiments, which also indicate that the assumptions made here and in the literature can be relaxed.