Mathematical Sciences: Fleming - Viot Processes with Selection

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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.