TY - JOUR

T1 - Fast non mean field networks: uniform in time averaging

AU - Barré, Julien

AU - Dobson, Paul

AU - Ottobre, Michela

AU - Zatorska, Ewelina

N1 - Funding Information:
The work of the authors was supported by the International Centre for Mathematical Sciences, Research-in-Groups programme. The work of the second author was supported by the Maxwell Institute Graduate School in Analysis and its Applications (MIGSAA), the Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016508/01, the Scottish Funding Council, the Heriot-Watt University, and the University of Edinburgh.
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021

Y1 - 2021

N2 - We study a population of N particles, which evolve according to a diffusion process and interact through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e., with O(N) particles, we consider the a priori more difficult case of a sparse network; that is, each particle interacts, on average, with O(1) particles. We also assume that the network's dynamics is much faster than the particles' dynamics, with the time-scale of the network described by a parameter ∊ > 0. We combine the averaging (∊ → 0) and the many-particles (N → ∞ ) limits and prove that the evolution of the particles' empirical density is described (after taking both limits) by a nonlinear Fokker-Planck equation; moreover, we give conditions under which such limits can be taken uniformly in time, hence providing a criterion under which the limiting nonlinear Fokker-Planck equation is a good approximation of the original system uniformly in time. The heart of our proof consists of controlling precisely the dependence on N of the averaging estimates.

AB - We study a population of N particles, which evolve according to a diffusion process and interact through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e., with O(N) particles, we consider the a priori more difficult case of a sparse network; that is, each particle interacts, on average, with O(1) particles. We also assume that the network's dynamics is much faster than the particles' dynamics, with the time-scale of the network described by a parameter ∊ > 0. We combine the averaging (∊ → 0) and the many-particles (N → ∞ ) limits and prove that the evolution of the particles' empirical density is described (after taking both limits) by a nonlinear Fokker-Planck equation; moreover, we give conditions under which such limits can be taken uniformly in time, hence providing a criterion under which the limiting nonlinear Fokker-Planck equation is a good approximation of the original system uniformly in time. The heart of our proof consists of controlling precisely the dependence on N of the averaging estimates.

KW - Averaging methods

KW - Interacting particle systems

KW - Markov semigroups

KW - Nonlinear Fokker-Plank equation

KW - Sparse interaction

UR - http://www.scopus.com/inward/record.url?scp=85103151137&partnerID=8YFLogxK

U2 - 10.1137/20M1328646

DO - 10.1137/20M1328646

M3 - Article

VL - 53

SP - 937

EP - 972

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 1

ER -