Article Dans Une Revue SIAM Journal on Mathematical Analysis Année : 2021

Fast Non-mean-field Networks: Uniform in Time Averaging

Résumé

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 timescale of the network described by a parameter ε > 0. We combine the averaging (ε → 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 non-linear Fokker-Planck equation; we moreover give conditions under which such limits can be taken uniformly in time, hence providing a criterion under which the limiting non-linear 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 in N of the averaging estimates.

Fichier principal
Vignette du fichier
2003.14230v2.pdf (411.57 Ko) Télécharger le fichier
Origine Fichiers éditeurs autorisés sur une archive ouverte
Licence

Dates et versions

hal-03889977 , version 1 (18-07-2025)

Licence

Identifiants

Citer

Julien Barré, Paul Dobson, Michela Ottobre, Ewelina Zatorska. Fast Non-mean-field Networks: Uniform in Time Averaging. SIAM Journal on Mathematical Analysis, 2021, 53 (1), pp.937-972. ⟨10.1137/20M1328646⟩. ⟨hal-03889977⟩
114 Consultations
78 Téléchargements

Altmetric

Partager

  • More