Research in Stochastic Processes

[4051052] lalley这个项目的主要重点将放在种群动力学的随机模型和相关的相互作用粒子系统上。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。研究的很大一部分将指向某些相互作用系统在非欧几里得图和其他非标准图上的行为。要调查的具体问题将包括竞争物种的共存、种群的生长和传播、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论领域有重要的联系，特别是渗透、首通道渗透和随机图论。该项目的第二个重点将是(1)研究有限和无限生成函数系统的分析技术，特别是在树状结构上随机游走和相关组合枚举问题中出现的分析技术；(2)混沌动力系统产生的时间序列的信号提取和推理过程。随机相互作用粒子系统作为模型被广泛应用于各个科学领域，特别是统计物理学和人口生物学，但也应用于经济学。这些系统由以各种方式相互作用的“粒子”（在不同的环境中可能代表分子、生物有机体、经济代理人等）组成，但通常以不确定的方式相互作用。这些系统的数学研究试图理解大规模的集体现象（如磁力、流行病在大群体中的传播、肿瘤生长等）是如何产生的，并由单个粒子相互作用的细节决定的。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结构如何影响宏观行为的本质，特别是理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群体中的社会和生物联系的不寻常几何结构中变化。

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.