Stochastic Spatial Processes

本研究涉及到相互作用粒子系统的理论与应用。这些过程是由相互作用的组件组成的大型系统的随机模型。这些系统模拟的现象包括：物种竞争、流行病、遗传性状的传播、催化化学反应等等。研究者将对几个具体问题进行研究。第一个项目涉及相互催化的分支过程，它模拟了两个种群的进化，它们的生长速度相互依赖。第二个项目涉及低密度相互作用粒子系统和测量值扩散之间的关系，特别是前者的重标版本收敛到超布朗运动。第三个项目处理数学遗传学中的一些问题，特别是二维整数格上的垫脚石模型的极限结果。在方案四中，考虑了一类广义分支随机漫步。解决这一类的局部灭绝与稳定性这一主要问题的关键似乎是分析具有一定最小分支率的特定情况，该分支率采取以前未考虑过的形式。本研究涉及到相互作用粒子系统或随机空间过程的理论与应用。本研究的目的是为了更好地定性地理解相互作用粒子系统所能很好地模拟的各种复杂现象。这些是由许多相互作用的组件组成的大型系统，通常带有随机或随机元素。例如，一个简单的遗传性状通过空间分布的种群进化的模型就符合这个框架。它结合了个体在空间分布的群体中的运动，以及小频率的突变。研究者希望表明，这种类型的模型可以比传统的、简单的、广泛使用的非空间模型给出更准确的结果。拟议研究的一部分涉及非常具体的模型和诸如此类的问题，而拟议研究的一部分将旨在开发处理这类模型的通用数学技术。

This research involves several topics in the theory and application of interacting particle systems. These processes are stochastic models for large systems of interacting components. Among the phenomena these systems model are: competition of species, epidemics, spread of genetic traits, catalytic chemical reactions, and more. The investigator will pursue research on several specific problems. The first project involves the mutually catalytic branching process, which models the evolution of two populations whose growth rates depend on one another. The second project concerns the relationship between low density interacting particle systems and measure-valued diffusions, especially the convergence of rescaled versions of the former to super-Brownian motion. The third project treats some questions from mathematical genetics, especially limiting results for the stepping stone model on the two dimensional integer lattice. In the fourth project a class of generalized branching random walks is considered. The key to resolving the main question of local extinction versus stability for this class appears to be the analysis of a particular case with a certain minimal branching rate which takes a form not previously considered. This research involves several topics in the theory and application of interacting particle systems or stochastic spatial processes. The goal of this research is to obtain a better qualitative understanding of various complex phenomena that interacting particle systems model well. These are large systems made up of many interacting components, usually with a stochastic or random element. For example, a simple model for the evolution of a genetic trait through a spatially distributed population fits into this framework. It incorporates the movement of individuals across spatially distributed colonies, and mutation at a small rate. The investigator hopes to show that this type of model can give more accurate results than the traditional, simpler non-spatial models widely used. Part of the proposed research concerns very specific models and questions such as this one, and part of the proposed research will aim at developing general mathematical techniques for handling models of this type.