Stochastic Spatial Processes

随机空间过程

• 批准号: 1208984
• 负责人: J. Theodore Cox
• 金额: $ 14万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1208984&HistoricalAwards=false
• 关键词: Stochastic; Spatial; Processes

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.

该项目是一类随机过程的研究，用于模拟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。通常，代理位于网络的节点处，无论是同质的，异构的还是随机的图，以及“信息”（感染，谣言，特征等）的传输。是随机的，但服从一些简单的局部规则。近年来，由于引入了“小世界”随机图作为万维网和互联网的模型，并试图分析大型社交网络，人们对这种类型的过程产生了兴趣。现在有一个快速增长的研究文献对这些新的模型，主要是数学以外的。 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交互机制，谣言或特征会在整个网络中迅速传播还是会很快消失？ 第二个相关的问题是：一个给定的系统是否会放松到一个准平衡状态，在很长一段时间内保持多样性。 对于一些“经典”的均匀晶格系统，这些问题有严格的结果，但分析较新的更异质的模型提出了重大的数学挑战。该项目的研究将确定这些模型的关键特征，即网络和相互作用机制的特征，这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提供一个严格的基础。更一般地说，该项目关注的是基于局部规则的大型随机系统如何随时间发展。研究的系统将从一个空间模型的系谱性状的进化在一个无限的人口位于均匀（地理）空间，连续和离散的，各种随机模型的异质随机图。 拟议中的研究将开发严格的数学方法，以确定基于启发式或平均场参数的预测是否有效。该研究将利用一系列数学技术，从分支过程，相互作用的粒子系统，渗流理论，有限马尔可夫链，随机游走和随机图。特别是该项目将利用，并进一步扩展，快速混合有限马尔可夫链的行为的结果。