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A new approach to the mean-field limit of Vlasov–Fokker–Planck equations

Didier Bresch, Pierre-Emmanuel Jabin and Juan Soler

Vol. 18 (2025), No. 4, 1037–1064
Abstract

We introduce a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov–Poisson–Fokker–Planck system for plasmas in dimension 2 together with a partial result in dimension 3. The method is broadly compatible with second-order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted Lp norms of the marginals or observables of the system, uniformly in the number of particles. This allows us to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in L2.

Keywords
kinetic equations, mean-field limit, Vlasov–Poisson–Fokker–Planck, marginals
Mathematical Subject Classification
Primary: 35Q70, 35Q83, 60H10, 70F45, 82D10
Milestones
Received: 7 March 2023
Revised: 21 September 2023
Accepted: 21 December 2023
Published: 27 March 2025
Authors
Didier Bresch
Laboratoire de Mathématiques UMR5127 CNRS
Université Savoie Mont Blanc
Le Bourget du Lac
France
Pierre-Emmanuel Jabin
Department of Mathematics
Pennsylvania State University
University Park, PA
United States
Juan Soler
Departamento de Matemática Aplicada
Facultad de Ciencias
Universidad de Granada
Granada
Spain

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