The large deviations of a multi-allele Wright-Fisher process mapped on the sphere

Abstract

This is the fourth in a series of papers devoted to the study of the large deviations of a Wright –Fisher process modeling the genetic evolution of a reproducing population.Variational considerations imply that if the process undergoes a large deviation, then it necessarily follows closely a definite path from its original to its current state. The favored paths were determined previously for a one-dimensional process subject to one-way mutation or natural selection, respectively, acting on a faster time scale than random genetic drift. The present paper deals with a general $d$-dimensional Wright–Fisher process in which any mutation or selection forces act on a time scale no faster than that of genetic drift. If the states of the process are represented as points on a $d$-sphere, then it can be shown that the position of a subcritically scaled process at a fixed “time” $T$ satisfies a large-deviation principle with rate function proportional to the square of the length of the great circle arc joining this position with the initial one (Hellinger–Bhattacharya distance). If a large deviation does occur, then the process follows with near certainty this arc at constant speed. The main technical problem circumvented is the degeneracy of the covariance matrix of the process at the boundary of the state space.