论文标题
在单模型Galton-Watson树上相互扩散的边际动力学
Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees
论文作者
论文摘要
考虑一个同质相互作用的扩散颗粒系统，该系统由单型galton-watson（UGW）树的节点标记，其中每个节点的状态像d二维扩散一样，其漂移系数取决于（其自身状态的历史），其自身的状态和状态的扩散系数仅依赖于（其自身的扩散系数）（其自身的扩散系数）（该历史上的历史）（该历史上的历史）（该历史上的历史）（这是一个历史上的历史）。在适当的规律性假设上，根据某个局部方程式，获得了典型节点邻域邻域动力学的边际分布的自主表征，这是一种新型的SDE，从McKean的意义上讲是非线性的。该方程式描述了有限维的非马克维亚随机过程，其演变在任何时候不仅取决于邻居的结构和当前状态，还取决于鉴于邻居节点状态的过去，当前状态的条件定律。这种边际分布引起了人们的关注，因为它们是边际分布的弱限制以及在许多稀疏随机图序列上相互作用扩散的经验度量，包括配置模型和ERDOS-RENYI图，其平均度收敛到有限的非零极限。结果在平均场状态中获得的补充经典结果，该结果表征了在完整图上均匀相互作用扩散的限制动力学，而在相应的非线性Markov过程中，节点的数量转移到了无穷大。但是，在稀疏图设置中，图的拓扑强大影响动力学，分析需要完全不同的方法。局部方程的存在和唯一性的证明依赖于粒子轨迹在UGW树上的精致新条件独立性和对称性，以及明智地使用度量变化。
Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton-Watson (UGW) tree, where the state of each node evolves like a d-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 SDE that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose 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. 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 Erdos-Renyi 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 UGW trees, as well as judicious use of changes of measure.
