Diffusions on Combinatorially Structured State Spaces

该项目涉及开发新的方法来研究具有组合结构状态空间的扩散，并受到群体遗传学和系统遗传学中出现的随机过程的推动。组合结构之所以出现在这些领域，是因为人们想要建模，例如，一个种群的基因组成如何随着时间的推移而变化，或者开发在系统发育树空间中有效移动的过程，以寻找最可能的一组生物。理解这些过程的挑战之一是确定组合结构的变化是如何随时间积累的。这个项目试图开发新的方法来理解这些过程，当它们的行为是由小变化的快速积累驱动的。研究生和本科生将参与这个项目。集合分区、整数分区和整数组合在组合随机过程的研究中起着核心作用。集分区和整数分区上的马尔可夫过程已经得到了很好的研究，但整数组合值过程的理论还没有得到很好的发展。本项目将开发研究整数组合值过程的扩散类似物的方法，即区间划分值过程，重点研究与无限多中性等位基因扩散模型的双参数扩展相关的过程。通过串珠破碎构造，利用区间分区和连续树之间的联系，本项目还将开发构造连续树值扩散的方法。该项目将发展这些过程的路径构建和分析特征，并建立不变性原则。本项目由概率计划和促进竞争研究的既定计划（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project involves developing new methods to study diffusions with combinatorially structured state-spaces and is motivated by stochastic processes that arise in population genetics and phylogenetics. Combinatorial structure arises in these areas because one would like to model, for example, how the genetic makeup of a population changes over time, or to develop processes that move efficiently through the space of phylogenetic trees in search of the most likely one for a group of organisms. One of the challenges in understanding these processes is determining how changes to the combinatorial structure accumulate over time. This project seeks to develop new methods for understanding such processes when their behavior is driven by the rapid accumulation of small changes. Graduate and undergraduate students will be engaged in this project.Set partitions, integer partitions, and integer compositions play a central role in the study of combinatorial stochastic processes. Markov processes on set partitions and integer partitions are well studied, but the theory for integer composition-valued processes is not as well developed. This project will develop methods for studying diffusion analogues of integer composition-valued processes, which are interval partition-valued processes, with a focus on processes related to the two-parameter extension of the infinitely-many-neutral-alleles diffusion model. Using the connection between interval-partitions and continuum trees through bead-crushing constructions, this project will also develop methods for constructing continuum-tree valued diffusions. This project will develop both pathwise constructions and analytic characterizations of these processes as well as establish invariance principles.This project is jointly funded by Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.