Nonparametric estimation for interacting particle systems: McKean-Vlasov models

• 成果类型: Article
• 署名作者: Della Maestra, Laetitia; Hoffmann, Marc
• 署名单位: Universite PSL; Universite Paris-Dauphine; Universite PSL; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01044-6
• 发表日期: 2022
• 页码: 551-613
• 关键词: DENSITY-ESTIMATION; division rate; inequalities; fluctuations; propagation; probability; selection; EQUATIONS; chaos; LAW

摘要

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 -> infinity. 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 Holder 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.