Stochastic models and their applications in population genetics

群体遗传学的一个重要目标是了解在群体中产生变异遗传景观的机制。在这一探索中提出了各种概率模型，包括离散马尔可夫链模型和连续扩散模型。连续模型通常是离散模型的近似，在计算上更为友好。本研究将重点研究泊松-狄利克雷分布、伊文斯抽样公式、弗莱明-维奥过程、重组模型、基因搭便车和凝聚等模型。许多这样的模型依赖于某些关键参数，如时间、种群大小、等位基因数量、突变率和选择适应度。不同的模型通过将相应的参数从一种方案移动到另一种方案来相互连接。本研究的目的之一是研究涉及这些参数的各种限制程序，并了解它们之间的关系。考虑一个种群，其中每个个体都属于特定的基因型。那么，类型频率就是类型空间上的一个概率度量。弗莱明-维奥过程是描述种群在突变和选择作用下类型频率进化的模型。加拿大数学家珀金斯（E. Perkins）的一个著名结论是，弗莱米格-维奥过程的研究可以从另一个具有丰富独立结构的超过程中得到。我的第二个目标是在帕金斯结果的基础上解决具有一般突变的弗莱明-维奥过程的大偏差问题。这一研究项目的成功必将揭示几种遗传模式之间的新关系和新结构。这对测度过程理论的研究及其在群体遗传学中的应用具有重要意义。

One important goal of population genetics is to understand the mechanisms that generate the variational genetic landscape in a population. A variety of probability models have been proposed in this exploration, including the discrete Markov chain models and the continuous diffusion models. The continuous model is usually an approximation to the discrete model and is more friendly in computation. This research will focus on models like Poisson-Dirichlet distribution, Ewens sampling formula, the Fleming-Viot process, recombination models, genetic hitchhiking, and the coalescent. Many such models depend on certain crucial parameters such as time, the population size, the number of alleles, mutation rate, and selective fitness. Different models are connected with each other by moving the corresponding parameters from one scheme to another. One objective of this research is to investigate various limiting procedures involving these parameters and understand their relations. Consider a population in which each individual belongs to a particular genotype. The type frequency is then a probability measure on the space of types.  A Fleming-Viot process is a model that describes the evolution of type frequency of a population under the influence of mutation and selection. A well known result by Canadian mathematician E. Perkins says that the study of a Flemig-Viot process can be obtained from another  superprocess that possesses rich independent structures. My second  objective is to solve the open problem of large deviations for Fleming-Viot process with general mutation based on Perkins' result.    The success of this research program will definitely reveal new relations and structures among several genetic models. It will also contribute greatly to the research in the theory of measure-valued process and their applications in population genetics.