Stochastic models and their applications in population genetics

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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.