Marginal dynamics of interacting diffusions on unimodular Galton–Watson trees
Marginal dynamics of interacting diffusions on unimodular Galton–Watson trees
复制标题
DOI：
10.1007/s00440-023-01226-4
复制
发表时间：
2020-09
影响因子：
2
通讯作者：
D. Lacker;K. Ramanan;Ruoyu Wu
中科院分区：
文献类型：
--
作者：
D. Lacker;K. Ramanan;Ruoyu Wu
考虑一个由单模高尔顿-沃森树的节点标记的均匀相互作用扩散粒子系统，其中每个节点的状态像ad维扩散一样无限小地演化，其漂移系数取决于其自身状态（的历史）和相邻节点的状态，并且其扩散系数仅取决于其自身状态（的历史）。在适当的系数正则性假设下，得到了典型节点邻域动力学的边缘分布在某个局部方程下的自治刻画，该局部方程是一类新的McKean意义下的非线性随机微分方程.该方程描述了一个有限维非马尔可夫随机过程，其无穷小演化在任何时候不仅取决于邻域的结构和当前状态，而且还取决于当前状态的条件律，该条件律给出了直到该时刻的相邻节点的状态的过去。这样的边际分布是有趣的，因为它们是许多稀疏随机图序列上的边际分布和相互作用扩散的经验测度的弱极限，包括配置模型和平均度收敛到有限非零极限的Erdös-Rényi图。得到的结果补充经典的结果在平均场制度，其特征在于在完全图上的均匀相互作用扩散的限制动力学，作为节点的数量趋于无穷大，在相应的非线性马尔可夫过程。然而，在稀疏图设置中，图的拓扑结构强烈地影响动力学，并且分析需要完全不同的方法。局部方程的存在性和唯一性的证明依赖于单模Galton-Watson树上粒子轨迹的微妙的新的条件独立性和对称性，以及明智地使用测量的变化。
Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton–Watson tree, where the state of each node evolves infinitesimally like ad-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on (the history of) its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for the marginal distribution of the dynamics of the neighborhood of a typical node in terms of a certain local equation, which is a new kind of stochastic differential equation that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the structure and current state of the neighborhood, but also on the conditional law of the current state given the past of the states of neighborhing nodes until that time. Such marginal distributions are of interest because they arise as weak limits of both marginal distributions and empirical measures of interacting diffusions on many sequences of sparse random graphs, including the configuration model and Erdös–Rényi graphs whose average degrees converge to a finite non-zero limit. The results obtained complement classical results in the mean-field regime, which characterize the limiting dynamics of homogeneous interacting diffusions on complete graphs, as the number of nodes goes to infinity, in terms of a corresponding nonlinear Markov process. However, in the sparse graph setting, the topology of the graph strongly influences the dynamics, and the analysis requires a completely different approach. The proofs of existence and uniqueness of the local equation rely on delicate new conditional independence and symmetry properties of particle trajectories on unimodular Galton–Watson trees, as well as judicious use of changes of measure.