Large sparse networks of interacting diffusions
Large sparse networks of interacting diffusions
复制标题
相互作用扩散的大型稀疏网络
DOI：
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Ruoyu Wu
中科院分区：
文献类型：
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作者：
D. Lacker;K. Ramanan;Ruoyu Wu
我们考虑一个大的稀疏的，可能是随机的，相互作用图$G_n$上的相互作用粒子系统，其中每个粒子像一个d维扩散一样无限小地演化，其漂移系数取决于其自身状态和相邻粒子的状态的历史，而扩散系数仅取决于其自身状态的历史。我们研究了当G_n的平均度几乎处处有界时这类粒子系统的极限。具体地说，在适当的系数和初始条件的假设下，我们证明了如果$G_n$在局部收敛意义下按分布收敛到局部有限图$G$，则相应的粒子动力学弱局部收敛到$G$上的某个极限扩散。我们还表明，对于某些图序列，包括稀疏的Erdos-Renyi图序列，经验测量的极限是确定性的，符合一个典型的粒子的法律，但在一般情况下，经验测量极限可能无法符合一个典型的粒子的法律，它甚至可以保持随机。此外，当$G$是么模Galton-Watson树时，在适当的假设下，我们用某个有限维非马尔可夫随机过程来刻画典型粒子邻域的极限动力学，该过程在任何时候的无穷小演化不仅取决于邻域的当前状态，而且还取决于给定到那时为止的邻近过程的过去的当前状态的条件定律。这解决了一个公开的问题，描述稀疏Erdos-Renyi图序列的极限动力学。重要成分的证据包括相关衰减估计，二阶马尔可夫随机场的粒子轨迹，和随机分析结果模仿伊藤过程。
We consider interacting particle systems on a large sparse, possibly random, interaction graph $G_n$, where each particle evolves infinitesimally like a d-dimensional diffusion whose drift coefficient depends on the histories of its own state and the states of neighboring particles, and the diffusion coefficient depends only on the history of its own state. We study limits of such particle systems in the case when the average degree of $G_n$ remains almost surely bounded. Specifically, under suitable assumptions on the coefficients and initial conditions, we show that if $G_n$ converges in distribution in the sense of local convergence to a locally finite graph $G$, then the corresponding particle dynamics converge weakly locally to a certain limit diffusion on $G$. We also show that for certain graph sequences, including sparse Erdos-Renyi graph sequences, the limit of the empirical measure is deterministic and coincides with the law of a typical particle; but show that in general, the empirical measure limit may fail to coincide with the law of a typical particle, and it could even remain stochastic. Furthermore, when $G$ is a unimodular Galton-Watson tree, under suitable assumptions we characterize the limiting dynamics of the neighborhood of a typical particle in terms of a certain finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the current state of the neighborhood, but also on the conditional law of the current state given the past of the neighborhood process until that time. This resolves the open problem of characterizing the limiting dynamics on sequences of sparse Erdos-Renyi graphs. Important ingredients of the proofs include correlation decay estimates, a second-order Markov random field property for particle trajectories, and a stochastic analytic result on mimicking Ito processes.
影响因子：
1.4
作者：
Detering, Nils;Fouque, Jean-Pierre;Ichiba, Tomoyuki
通讯作者：
Ichiba, Tomoyuki