SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 937-972, January 2021.
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 $\varepsilon>0$. We combine the averaging ($\varepsilon \rightarrow 0$) and the many-particles ($N \rightarrow \infty$) 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.

中文翻译：

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快速的非均值网络：时间平均一致

SIAM数学分析杂志，第53卷，第1期，第937-972页，2021年1月。
我们研究了$ N $粒子的数量，这些粒子根据扩散过程进行演化并通过动态网络进行交互。反过来，网络的演化与粒子的位置相关。与每个粒子与每个其他粒子（即与$ O（N）$粒子）相互作用的平均场方式相反，我们认为稀疏网络的先验条件更为困难；也就是说，每个粒子平均与$ O（1）$粒子相互作用。我们还假设网络的动力学比粒子的动力学快得多，网络的时标由参数$ \ varepsilon> 0 $来描述。我们结合了平均值（$ \ varepsilon \ rightarrow 0 $）和多粒子（$ N \ rightarrow \ infty $）的极限，证明了粒子的演化 经验密度由非线性Fokker-Planck方程描述（同时取两个极限）；此外，我们给出了可以在时间上均匀地采用这些极限的条件，因此提供了一个准则，在该条件下极限非线性Fokker-Planck方程在时间上均匀地近似于原始系统。我们证明的核心在于精确控制平均估计对$ N $的依赖性。