Stochastic Spatial Processes

这个建议将解决相互作用粒子系统和随机空间过程理论中的问题。 这些随机过程是具有许多相互作用的“组件”（细胞，个体，粒子，植物等）的大型系统的模型。 这些系统模型的一些现象的例子是：物种竞争，流行病，人口增长，遗传性状的进化。这一领域研究的一个主要目标是了解大系统的宏观行为如何依赖于组件之间的个体相互作用。 两个竞争模型将被研究，空间随机Lotka-Volterra模型和多类型接触过程。 对于Lotka-Volterra模型，目标是确定对应于一个物种生存和两个物种共存的参数区域。该项目是研究人员正在进行的努力的一部分，以了解和利用相互作用的粒子系统和测量值扩散之间的标度关系。对于多型接触过程，感兴趣的问题是关于规则树的生存与灭绝。研究者期望发现新的相依赖的生存和灭绝现象，不存在于晶格的情况下。该建议的第二个一般领域涉及群体遗传学引起的问题。地理结构种群中遗传性状的进化通常采用木村的垫脚石模型（英语：Stepping stone model）来建模，其中关于亲属关系的度量的问题被重新表述为关于合并随机游走系统行为的问题。 研究者以前的研究包括这些系统的极限定理的工作，大型，二维格集环面或环绕边界条件。 这里的目标是显示这些定理在广泛的边界条件下的鲁棒性，证明它们在群体遗传学中的应用是合理的。 这些随机过程是具有许多相互作用的组件（细胞，个体，粒子，植物等）的大型系统的模型。 本研究的目标是更好地定性了解相互作用粒子系统模型的各种复杂现象，例如：物种竞争，流行病，人口增长，遗传性状进化。 这一领域研究的一个主要目标是了解大系统的宏观行为如何依赖于组件之间的个体相互作用。几个具体的模型将进行研究，包括一个众所周知的物种之间的竞争模型的随机空间版本。除了工作在特定的模型，调查人员将试图延长的有效性，为一些特定的模型建立一些近似定理，以处理更一般的，从而证明他们在应用中的使用。

This proposal will address questions in the theory of interacting particle systems and stochastic spatial processes. These stochastic processes are models for large systems with many interacting ``components'' (cells, individuals, particles, plants, etc.). Some examples of the phenomena these systems model are: competition of species, epidemics, population growth, evolution of genetic traits. A principal goal of research in this area is to understand how the macroscopic behavior of large systems depends on the individual interactions between components. Two competition models will be studied, a spatial stochastic Lotka-Volterra model and a multitype contact process. For the Lotka-Volterra model, the objectives are to determine the parameter regions which correspond to survival of one species and coexistence of both species. The project is part of the investigator's ongoing efforts to understand and exploit the scaling relationship between interacting particle systems and measure-valued diffusions. For the multitype contact process, the questions of interest concern survival versus extinction on regular trees. The investigator expects to find new phase dependent survival and extinction phenomena that do not exist in the lattice case. A second general area of this proposal addresses problems motivated by population genetics. The evolution of genetic traits in geographically structured populations is often modeled with Kimura's stepping stone model, where questions about measures of kinship are reformulated in terms of questions about the behavior of coalescing random walk systems. The investigator's previous research includes work on limit theorems for these systems on large, two-dimensional lattice sets with torus or wrap-around boundary conditions. The goal here is to show robustness of these theorems over a wide range of boundary conditions, justifying their use in population genetics.This proposal involves research in the theory of interacting particle systems and stochastic spatial processes. These stochastic processes are models for large systems with many interacting components (cells, individuals, particles, plants, etc.). The goal of this research is to obtain a better qualitative understanding of various complex phenomena that interacting particles systems model well, such as models of: competition of species, epidemics, population growth, evolution of genetic traits. A principal goal of research in this area is to understand how the macroscopic behavior of large systems depends on the individual interactions between components. Several specific models will be studied, including a stochastic spatial version of a well known model for competition between species. In addition to work on specific models, the investigator will try to extend the validity of some approximation theorems established for some specific models to handle more general ones, thus justifying their use in applications.