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
SN - 0036-1410
VL - 53
SP - 937
EP - 972
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
IS - 1
ER -