Stochastic Spatial Processes

随机空间过程

• 批准号: 0204422
• 负责人: J. Theodore Cox
• 金额: $ 12.3万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204422&HistoricalAwards=false
• 关键词: Stochastic; Spatial; Processes

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.

这项研究涉及相互作用粒子系统的理论和应用的几个主题。这些过程是由相互作用的组件组成的大系统的随机模型。这些系统模型中的现象包括：物种竞争、流行病、遗传特征的传播、催化化学反应等。调查员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枝过程，该过程模拟了两个增长速度相互依赖的种群的进化。第二个项目涉及低密度相互作用的粒子系统和测值扩散之间的关系，特别是前者的重新标度版本到超布朗运动的收敛。第三个项目讨论了数学遗传学中的一些问题，特别是对二维整数格上的垫脚石模型的有限结果。第四个方案考虑了一类广义分枝随机游动。解决这一类局部灭绝与稳定性的主要问题的关键似乎是分析具有某一最小分支率的特定情况，该分支率采取以前没有考虑过的形式。这项研究涉及相互作用的粒子系统或随机空间过程的理论和应用的几个主题。这项研究的目的是为了更好地定性地理解各种相互作用的粒子系统很好地模拟的复杂现象。这些是由许多相互作用的组件组成的大系统，通常带有随机或随机元素。例如，一个简单的遗传特征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移动，并以很小的速度发生突变。研究人员希望表明，这种类型的模型可以给出比传统的、更简单的、广泛使用的非空间模型更准确的结果。部分拟议的研究涉及非常具体的模型和问题，如这一类，而部分拟议的研究将旨在开发处理这类模型的一般数学技术。