Tracing the Path of a Wright-Fisher Process with One-way Mutation in the Case of a Large Deviation

Abstract

The large deviations theory developed by Wentzell for discrete time Markov chains is used to show that if the state of a Wright-Fisher process modeling the frequency of an allele in a biological population undergoes a transition from one value to another over a number of generations which is large but much smaller than the size of the population, then there is a preferred path which the process follows closely with near certainty in the intervening time. This path was identified in [8]_in the case in which there is only random drift acting on the genes. The case in which one-way mutation is added to the drift was cursorily mentioned in [8]_and is fully treated here. The preferred path in this case is shown to be an exponential, a parabola, a hyperbolic cosine or a trigonometric cosine, depending on the mutation parameter and the boundary conditions involved.