快速非平均场网络：时间平均一致
Fast non mean-field networks: uniform in time averaging
摘要：
我们研究的人口的$N$粒子，根据扩散过程和相互作用，通过一个动态网络。反过来，网络的演化与粒子的位置相耦合。与平均场机制相反，在平均场机制中，每个粒子与每一个其他粒子相互作用，即与$O(N)$粒子相互作用，我们考虑稀疏网络的先验更困难的情况;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，网络的时间尺度由参数$ε>0$描述。我们把平均值（$ε\rightarrow 0$）和许多颗粒（$N \rightarrow \infty$）极限，并证明了粒子经验密度的演化是描述的（在取两个极限之后）通过非线性Fokker-Planck方程;我们还给出了这样的极限在时间上一致的条件，因此提供了一个标准，在该标准下，极限非线性Fokker-Planck方程是原始系统在时间上一致的良好近似。我们证明的核心是精确控制平均估计值对$N$的依赖性。
英文摘要：
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 $ε>0$. We combine the averaging ($ε\rightarrow 0$) 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 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.
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