Large deviations for singular and degenerate diffusion models in adaptive evolution

Abstract

In the course of Darwinian evolution of a population, punctualism is an important phenomenon whereby long periods of genetic stasis alternate with short periods of rapid evolutionary change. This paper provides a mathematical interpretation of punctualism as a sequence of change of basin of attraction for a diffusion model of the theory of adaptive dynamics. Such results rely on large deviation estimates for the diffusion process. The main difficulty lies in the fact that this diffusion process has degenerate and non-Lipschitz diffusion part at isolated points of the space and non-continuous drift part at the same points. Nevertheless, we are able to prove strong existence and the strong Markov property for these diffusions, and to give conditions under which pathwise uniqueness holds. Next, we prove a large deviation principle involving a rate function which has not the standard form of diffusions with small noise, due to the specific singularities of the model. Finally, this result is used to obtain asymptotic estimates for the time needed to exit an attracting domain, and to identify the points where this exit is more likely to occur.