Weak quantitative propagation of chaos via differential calculus on the space of measures

Abstract

Consider the metric space $({\mathcal{P}}_2({\mathbb{R}}^{\mathit{d}}),{\mathit{W}}_2)$ of square integrable laws on ${\mathbb{R}}^{\mathit{d}}$ with the topology induced by the 2-Wasserstein distance ${\mathit{W}}_2$. Let $\mathrm{\Phi }:{\mathcal{P}}_2({\mathbb{R}}^{\mathit{d}})\to \mathbb{R}$ and $\mathit{\mu }\in {\mathcal{P}}_2({\mathbb{R}}^{\mathit{d}})$. In this work, we consider (a) ${\mathit{\mu }}_{\mathit{N}}$ being the empirical measure of N-samples from μ, and the other case in which (b) ${\mathit{\mu }}_{\mathit{N}}$ is the empirical measure of marginal laws of the particle system of a McKean–Vlasov PDE ${({\mathit{\mu }}_{\mathit{t}})}_{\mathit{t}}$. The main result of this paper is to show that under suitable regularity conditions, we have

$$|\mathrm{\Phi }(\mathit{\mu })-\mathbb{E}\mathrm{\Phi }({\mathit{\mu }}_{\mathit{N}})|=\sum _{\mathit{j}=1}^{\mathit{k}-1}\frac{{\mathit{C}}_{\mathit{j}}}{{\mathit{N}}^{\mathit{j}}}+\mathit{O}\left(\frac{1}{{\mathit{N}}^{\mathit{k}}}\right),$$

for some positive constants ${\mathit{C}}_1,\dots ,{\mathit{C}}_{\mathit{k}-1}$ that do not depend on N, where k corresponds to the degree of smoothness. The case where the samples are i.i.d. is studied using functional derivatives on the space of measures. The case of particle systems relies on an Itô-type formula for the flow of probability measures and is intimately connected to PDEs on the space of measures, called the master equation in the literature of mean-field games. We state general regularity conditions required for each case and analyze the regularity in the case of functionals of the laws of McKean–Vlasov PDEs. Ultimately, this work reveals quantitative estimates of propagation of chaos for interacting particle systems. Furthermore, we are able to provide weak propagation of chaos estimates for ensembles of interacting particles and show that these may have some remarkable properties.