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

DOI10.1137/20M1328646zbMATH Open1469.60302arXiv2003.14230MaRDI QIDQ5150325

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Publication date: 15 February 2021

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Abstract: 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 $epsilon>0$. We combine the averaging ($epsilonightarrow0$) and the many particles ($Nightarrowinfty$) 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.

Full work available at URL: https://arxiv.org/abs/2003.14230

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