Wright-Fisher Diffusion in One Dimension

TL;DR

This paper studies the mathematical properties of Wright-Fisher diffusion processes in one dimension, focusing on boundary behavior, regularity, and asymptotic solutions relevant to population genetics.

Contribution

It provides a detailed regularity theory and asymptotic analysis for Wright-Fisher type equations with degenerate operators on [0, 1].

Findings

Sharp regularity results for zero flux boundary conditions

Precise asymptotics of solutions as time approaches 0 and infinity

Asymptotic behavior near the endpoints 0 and 1

Abstract

We analyze the diffusion processes associated to equations of Wright-Fisher type in one spatial dimension. These are defined by a degenerate second order operator on the interval [0, 1], where the coefficient of the second order term vanishes simply at the endpoints, and the first order term is an inward-pointing vector field. We consider various aspects of this problem, motivated by applications in population genetics, including a sharp regularity theory for the zero flux boundary conditions, as well as a derivation of the precise asymptotics for solutions of this equation, both as t goes to 0 and infinity, and as x goes to 0, 1.