Research in Stochastic Processes

随机过程研究

• 批准号: 0405102
• 负责人: Steven Lalley
• 金额: $ 26.7万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405102&HistoricalAwards=false
• 关键词: Research; Stochastic; Processes

0405102Lalley The principal focus of the project will be on stochastic models of population dynamics and related interacting particle systems. These include models of spatial growth and competition, and models of epidemics and endemics in geographically structured populations. A substantial part of the research will be directed to the behavior of certain interacting systems on noneuclidean and other nonstandard graphs. Particular problems to be investigated will include coexistence of competing species, growth and spread of populations, weak-to-strong survival transitions, and spatial propagation of epidemics and endemics. These have important connections with other developing areas of probability, especially percolation, first-passage percolation, and random graph theory. Secondary foci of the project will be on (1) analytic techniques for studying systems of generating functions, both finite and infinite, especially those that arise in random walk and related combinatorial enumeration problems on tree-like structures; and (2) signal extraction and inference procedures for time series generated by chaotic dynamical systems. Stochastic interacting particle systems are widely used as models in various areas of science, especially in statistical physics and in population biology, but also in economics. These systems consist of "particles" (which in different contexts may represent molecules, biological organisms, economic agents, etc.) that interact in various ways, but generally in a non-deterministic fashion. Mathematical studies of such systems seek to understand how large-scale collective phenomena (such as magnetism, propagation of epidemics in large populations, tumor growth, etc.) arise from and are determined by the details of the individual particle interactions. An important goal of this project is to contribute to our understanding of how the geometry describing the interactions between particles affects the nature of macroscopic behavior, and in particular, to understand how the behavior of certain model particle systems may vary in unusual geometries that may describe social and biological connections in certain animal and human populations.

0405102 Lalley该项目的主要重点将是人口动态和相关的相互作用粒子系统的随机模型。这些模型包括空间增长和竞争模型，以及地理结构人口中的流行病和地方病模型。研究的一个重要部分将是针对某些相互作用的系统在非欧几里德和其他非标准图的行为。特别要调查的问题将包括竞争物种的共存，人口的增长和蔓延，弱到强的生存过渡，以及流行病和地方病的空间传播。这些与概率论的其他发展领域有着重要的联系，特别是渗流、第一通道渗流和随机图论。该项目的第二个重点将是（1）分析技术研究系统的生成函数，有限和无限的，特别是那些出现在随机行走和相关的组合枚举问题的树状结构;和（2）信号提取和推理程序的时间序列产生的混沌动力系统。 随机相互作用粒子系统被广泛用作各个科学领域的模型，特别是在统计物理学和人口生物学中，但也在经济学中。这些系统由“粒子”组成（在不同的语境中可能代表分子、生物有机体、经济主体等）。以各种方式相互作用，但通常以非确定性的方式。对这类系统的数学研究试图理解大规模的集体现象（如磁力、流行病在大量人群中的传播、肿瘤生长等）由单个粒子相互作用的细节产生并决定。该项目的一个重要目标是帮助我们理解描述粒子之间相互作用的几何形状如何影响宏观行为的性质，特别是理解某些模型粒子系统的行为如何在不寻常的几何形状中变化，这些几何形状可能描述某些动物和人类群体中的社会和生物联系。