Branching Brownian motion and population models

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In recent years, there has been growing interest in using tools from probability theory to solve biological problems. Probability theory has contributed substantially to the field of population genetics because many of the factors that are responsible for the evolution of populations are best modeled as random events. The aim of this project is to make further contributions in this area by undertaking a detailed mathematical study of some natural models of randomly evolving populations.The first part of the project focuses on branching Brownian motion with absorption. Particles move according to Brownian motion with drift, divide into two at a constant rate, and are absorbed upon hitting the origin. This process can be viewed as modeling a population undergoing selection. The particle positions represent fitnesses of individuals, the branching events represent births, and absorption at the origin models the death of unfit individuals. The goal will be to provide sharp estimates for the probability that the process survives for a long time and for the position of the right-most particle. The second part of the project involves studying a model of a population of fixed size in which individuals acquire beneficial mutations. The goal will be to prove results concerning the speed of evolution, the distribution of the fitnesses of individuals in the population, and the genealogy of the population. The third part of the project involves further study of the beta coalescent, a process which can be used to describe the genealogy of populations with large family sizes.