Mathematical Sciences: Fleming - Viot Processes with Selection

Fleming-Viot(FV)过程是概率度量值扩散过程，它对种群中类型分布的演化进行建模。这项建议涉及与无限多等位基因选择模型有关的三个主题。第一个主题涉及带有选择的最简单的FV过程，目的是获得转移函数的显式公式(尽管没有中性情况下那么显式)。第二个主题涉及Tchida介绍的一个模型(他称之为近中性突变模型)，该模型可以表示为具有无界选择强度函数的FV过程。目的是建立模型的唯一性，并研究模型的遍历性质，以期解释Tchida的一些模拟结果。第三个主题涉及高形和尼尔引入的等位基因系谱的概念。人们可以制定一个FV过程，在无限多的等位基因设置中跟踪这样的家谱。在中性情况下有几个未决的问题，但挑战是以更严格的方式分析选择的模型。这一建议涉及到群体遗传学中的某些随机(即随机)过程。种群中的每个个体都被认为是某种基因“类型”。这些过程模拟了种群中不同类型频率的演变，考虑了随机交配、突变和自然选择等遗传因素。让一组可能的类型是任意的(特别是无限的)，就可以对“无限多个等位基因”这一假设进行建模。它还允许人们将历史或家谱信息合并到对个人类型的描述中。三个问题，涉及三个具体的模型(每个来自生物学文献)，被提出。这些问题涉及到为模型提供更好的数学理解，并可能解释生物学家通过经验发现的某些现象。每一种模式都涉及选择，这使问题变得相当复杂，但与简单的中性情况相比，它更具吸引力。选择的困难在于，人们不能获得关于感兴趣的量的闭合线性方程组。尽管如此，还是提出了几个有助于克服这一问题的想法。

Fleming--Viot (FV) processes are probability-measure-valued diffusion processes that model the evolution of the distribution of types in a population. This proposal concerns three topics related to the infinitely-many-alleles model with selection. The first topic involves the simplest FV process with selection, and the aim is to obtain an explicit formula for the transition function (albeit less explicit than in the neutral case). The second topic concerns a model introduced by Tachida (referred to by him as a nearly neutral mutation model), which can be formulated as an FV process with an unbounded selection intensity function. The aim is establish uniqueness and study the ergodic properties of the model, with the hope of explaining some of Tachida's simulation findings. The third topic involves the concept of allelic genealogy introduced by Takahata and Nei. One can formulate an FV process that keeps track of such genealogies in the infinitely-many-alleles setting. There are several open problems in the neutral case, but the challenge is to analyze the selective models in a more rigorous way. This proposal is concerned with certain stochastic (i.e., random) processes in population genetics. Each individual in a population is assumed to be of some genetic "type." The processes model the evolution of the frequencies of the various types in the population, taking into account such genetic factors as random mating, mutation, and natural selection. Letting the set of possible types be arbitrary (in particular, infinite) allows one to model the ``infinitely many alleles'' assumption. It also allows one to incorporate historical or genealogical information into the description of an individual's type. Three problems, concerning three specific models (each from the biological literature), are proposed. The problems are concerned with providing a better mathematical understanding of the models and possibly explaining certain phenomena discovered empirically by biologists. Each model involves selection, which complicates it considerably, but makes it of greater interest than simply the neutral case would be. The difficulty with selection is that one cannot obtain closed systems of linear equations for the quantities of interest. Nevertheless, several ideas are proposed that should help to overcome this.