Stochastic Spatial Processes

该项目是对一类随机过程的研究，用于为具有许多相互作用的“代理”（细胞、个体、组件、粒子、植物）的大型系统建模。通常，智能体位于网络的节点上，可以是同质图、异质图或随机图，智能体之间的“信息”（感染、谣言、特征等）传递是随机的，但遵循一些简单的局部规则。近年来，“小世界”随机图被用作万维网和互联网的模型，并试图分析大型社交网络，这激发了人们对这类过程的兴趣。现在，关于这些新模型的研究文献正在迅速增长，主要是在数学之外。这篇文献中考虑的问题类型的一个例子是：给定一个特定的网络和互动机制，谣言或特征会在网络中迅速传播还是会迅速消亡？第二个相关的问题是：一个给定的系统是否会放松到一种准平衡状态，并在很长一段时间内保持多样性？对于一些“经典的”同质晶格系统，这些问题有严格的结果，但是分析更新的更异构的模型提出了重大的数学挑战。本项目的研究将确定这些模型的关键特征，即网络特征和相互作用机制的特征，这些特征将决定它们的长期行为，并开发严格的数学方法来验证相应的预测。该项目将为进一步的研究奠定坚实的基础。更一般地说，该项目关注的是基于本地规则的大型随机系统如何随着时间的推移而发展。所研究的系统将从位于同质（地理）空间（包括连续和离散）的无限种群中家谱特征进化的空间模型到异构随机图上的各种随机模型。拟议的研究将发展严格的数学方法，以确定基于启发式或平均场参数的预测是否有效。该研究将利用分支过程、相互作用粒子系统、渗透理论、有限马尔可夫链、随机漫步和随机图等一系列数学技术。特别是该项目将利用并进一步推广关于快速混合有限马尔可夫链行为的结果。

The project is a study of a class of stochastic processes used to model large systems with many interacting "agents" (cells, individuals, components, particles, plants). Typically the agents are located at the nodes of a network, either a homogeneous, heterogeneous or random graph, and transmission of ``information'' (infection, rumor, traits, etc.) between agents is random but obeys some simple local rule. Interest in processes of this type has been spurred in recent years by the introduction of ``small world'' random graphs used as models of the world wide web and the internet, and attempts to analyze large social networks. There is now a rapidly growing research literature on these new models, primarily outside of mathematics. An example of the type of question considered in this literature is: given a particular network and interaction mechanism, will a rumor or trait spread rapidly throughout the network or will it quickly die out? A second, related question is: will a given system relax into a quasi-equilibrium which maintains diversity for a very long time. There are rigorous results available for such questions for some ``classic'' homogeneous lattice systems, but analyzing the newer more heterogeneous models raises significant mathematical challenges. The research in this project will identify key features of these models, i.e. features of the network and of the interaction mechanism, which will determine their long-term behavior, and develop rigorous mathematical methods to validate corresponding predictions. The project will provide a rigorous foundation from which further research can be based.More generally, the project is concerned with how large stochastic systems based on local rules develop over time. Systems studied will vary from a spatial model for the evolution of genealogical traits in an infinite population located in homogeneous (geographic) space, both continuous and discrete, to a variety of stochastic models on heterogeneous random graphs. The proposed research will develop rigorous mathematical methodologies for determining whether or not predictions made based on heuristic or meanfield arguments are valid. The research will make use of a range of mathematical techniques from branching processes, interacting particle systems, percolation theory, finite Markov chains, random walks, and random graphs. In particular the project will make use of, and further extend, results on the behavior of rapidly mixing finite Markov chains.