We consider an interacting N-particle system with the vision geometrical constraints and reflected noises, proposed as a model for collective behavior of individuals. We rigorously derive a continuity-type of mean-field equation with discontinuous kernels and the normal reflecting boundary conditions from that stochastic particle system as the number of particles N goes to infinity. More precisely, we provide a quantitative estimate of the convergence in law of the empirical measure associated to the particle system to a probability measure which possesses a density which is a weak solution to the continuity equation. This extends previous results on an interacting particle system with bounded and Lipschitz continuous drift terms and normal reflecting boundary conditions by Sznitman [J. Funct. Anal. 56 (1984) 311-336] to that one with discontinuous kernels.
|Number of pages||36|
|Journal||Mathematical Models and Methods in Applied Sciences|
|Publication status||Published - 2018 Feb 1|
Bibliographical noteFunding Information:
Y.P.C. was partially supported by EPSRC Grant EP/K008404/1, ERC-Starting Grant HDSPCONTR “High-Dimensional Sparse Optimal Control”, and Alexander Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researcher. Y.P.C. is also supported by National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (Nos. 2017R1C1B2012918 and 2017R1A4A1014735). The authors warmly thank Professor Maxime Hauray for helpful discussion and valuable comments. The authors also acknowledge the Institut Mittag-Leﬄer, and particularly Professor José A. Carrillo, where this work was partially done.
All Science Journal Classification (ASJC) codes
- Modelling and Simulation
- Applied Mathematics