论文标题
快速的非平均场网络：时间平均
Fast non mean-field networks: uniform in time averaging
论文作者
论文摘要
我们研究了$ n $颗粒的人群，这些颗粒是根据扩散过程进化的，并通过动态网络相互作用。反过来，网络的演变耦合到粒子的位置。与平均场状态相比，每个粒子都与其他粒子相互作用，即与$ o（n）$粒子相互作用，我们认为稀疏网络的先验案例更加困难。也就是说，每个粒子平均与$ O（1）$粒子相互作用。我们还假设网络的动力学比粒子的动力学快得多，而网络的时间尺度由参数$ε> 0 $描述。我们结合了平均（$ε\ rightarrow 0 $）和许多粒子（$ n \ rightarrow \ infty $）限制，并证明了粒子的经验密度的演变（在两种限制之后）通过非线性的fokker-planck equation描述；此外，我们提供了可以及时统一采用此类限制的条件，因此提供了一个标准，在该标准下，限制了非线性fokker-Planck方程是对原始系统的良好近似值。我们证明的核心包括准确控制平均估计值的$ n $依赖性。
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.
