Research in Stochastic Processes

该项目的主要重点将是种群动力学的随机模型和相关的相互作用的粒子系统。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。很大一部分研究将针对非欧几里德和其他非标准图上的某些相互作用系统的行为。要调查的具体问题将包括竞争物种的共存、种群的增长和传播、从弱到强的生存过渡以及流行病和地方病的空间传播。这些与其他发展中的概率领域有重要的联系，特别是渗流、首次通过渗流和随机图论。该项目的次要重点将是(1)研究有限和无限母函数系统的分析技术，特别是在树状结构上的随机游动和相关的组合计数问题中出现的函数系统；以及(2)混沌动力系统产生的时间序列的信号提取和推理过程。随机相互作用的粒子系统被广泛应用于科学的各个领域，特别是在统计物理和人口生物学中，而且在经济学中也有广泛的应用。这些系统由“粒子”组成(在不同的上下文中，粒子可能代表分子、生物有机体、经济代理人等)。它们以不同的方式相互作用，但通常是以一种非确定性的方式。对这类系统的数学研究试图理解大规模集体现象(如磁性、流行病在大人口中的传播、肿瘤生长等)是如何发生的。产生于单个粒子相互作用的细节，并由其决定。这个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如何影响宏观行为的性质，特别是了解某些模型粒子系统的行为如何在可能描述某些动物和人类种群中的社会和生物联系的不寻常几何中发生变化。

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.