A law of large numbers and large deviations for interacting diffusions on Erdos-Renyi graphs
A law of large numbers and large deviations for interacting diffusions on Erdos-Renyi graphs
复制标题
DOI：
10.1142/s0219493720500100
复制
发表时间：
2020-04-01
影响因子：
1.1
通讯作者：
Giacomin, Giambattista
中科院分区：
文献类型：
--
作者：
Coppini, Fabio;Dietert, Helge;Giacomin, Giambattista
我们考虑一类由微分方程（随机和确定性）描述的粒子系统，其中相互作用网络由一个Erdos-Renyi图的实现决定，参数p(n)是（0,1）的元素，其中n是图的大小（即粒子的数量）。如果p(n) 1，则图是完全图（平均场模型），并且众所周知，在适当的假设下，经验测度收敛为n- >∞，收敛于PDE的解：随机情况下的McKean-Vlasov（或Fokker-Planck）方程，或确定性情况下的Vlasov方程。已经证明，这适用于相当一般的相互作用网络，包括具有lim(n) p(n)n =∞的Erdes-Renyi图，并适当地重新密封相互作用以解释p(n)引入的稀释。然而，这些结果是在对初始数据的强假设下证明的，初始数据必须是混沌的，即一系列独立的同分布随机变量。我们贡献的目的是提出结果-大数定律和大偏差原理-只假设初始条件的经验测量的收敛性。
We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdos-Renyi graph with parameter p(n) is an element of (0, 1], where n is the size of the graph (i.e. the number of particles). If p(n) 1, the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as n -> infinity to the solution of a PDE: a McKean-Vlasov (or Fokker-Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdes-Renyi graphs with lim(n) p(n)n = infinity, and properly resealing the interaction to account for the dilution introduced by p(n). However, these results have been proven under strong assumptions on the initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results - Law of Large Numbers and Large Deviation Principle - assuming only the convergence of the empirical measure of the initial condition.