Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

Young Pil Choi; Samir Salem

doi:10.1142/S0218202518500070

Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

Young Pil Choi, Samir Salem

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	223-258
Number of pages	36
Journal	Mathematical Models and Methods in Applied Sciences
Volume	28
Issue number	2
DOIs	https://doi.org/10.1142/S0218202518500070
Publication status	Published - 2018 Feb 1

Bibliographical note

Funding 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.

Publisher Copyright:
© 2018 World Scientific Publishing Company.

All Science Journal Classification (ASJC) codes

Modelling and Simulation
Applied Mathematics

Access to Document

10.1142/S0218202518500070

Cite this

@article{b156fbe1139b4eb3b457b696caf99d03,

title = "Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones",

abstract = "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.",

author = "Choi, {Young Pil} and Samir Salem",

note = "Funding 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 Jos{\'e} A. Carrillo, where this work was partially done. Publisher Copyright: {\textcopyright} 2018 World Scientific Publishing Company.",

year = "2018",

month = feb,

day = "1",

doi = "10.1142/S0218202518500070",

language = "English",

volume = "28",

pages = "223--258",

journal = "Mathematical Models and Methods in Applied Sciences",

issn = "0218-2025",

publisher = "World Scientific Publishing Co. Pte Ltd",

number = "2",

}

TY - JOUR

T1 - Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

AU - Choi, Young Pil

AU - Salem, Samir

N1 - Funding 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. Publisher Copyright: © 2018 World Scientific Publishing Company.

PY - 2018/2/1

Y1 - 2018/2/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85038915029&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038915029&partnerID=8YFLogxK

U2 - 10.1142/S0218202518500070

DO - 10.1142/S0218202518500070

M3 - Article

AN - SCOPUS:85038915029

SN - 0218-2025

VL - 28

SP - 223

EP - 258

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

IS - 2

ER -

Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

Abstract

Bibliographical note

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this