Nonparametric estimation for interacting particle systems: McKean–Vlasov models

We consider a system of N interacting particles, governed by transport and diffusion, that converges in a mean-field limit to the solution of a McKean–Vlasov equation. From the observation of a trajectory of the system over a fixed time horizon, we investigate nonparametric estimation of the solution of the associated nonlinear Fokker–Planck equation, together with the drift term that controls the interactions, in a large population limit N→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document}. We build data-driven kernel estimators and establish oracle inequalities, following Lepski’s principle. Our results are based on a new Bernstein concentration inequality in McKean–Vlasov models for the empirical measure around its mean, possibly of independent interest. We obtain adaptive estimators over anisotropic Hölder smoothness classes built upon the solution map of the Fokker–Planck equation, and prove their optimality in a minimax sense. In the specific case of the Vlasov model, we derive an estimator of the interaction potential and establish its consistency.