Excursion theory for the Wright-Fisher diffusion

TL;DR

This paper develops a specialized excursion theory for the Wright-Fisher diffusion with mutation, enabling detailed analysis of its boundary behaviors and pathwise construction.

Contribution

It introduces a novel excursion framework for Wright-Fisher diffusion with mutation, bridging classical and general excursion theories.

Findings

Constructed a boundary-specific excursion theory for Wright-Fisher diffusion.

Developed a Poisson process of killed paths for pathwise construction.

Provided insights into boundary behaviors of the diffusion process.

Abstract

In this work, we develop excursion theory for the Wright--Fisher diffusion with mutation. Our construction is intermediate between the classical excursion theory where all excursions begin and end at a single point and the more general approach considering excursions of processes from general sets. Since the Wright--Fisher diffusion has two boundary points, it is natural to construct excursions which start from a specified boundary point, and end at one of two boundary points which determine the next starting point. In order to do this we study the killed Wright--Fisher diffusion, which is sent to a cemetery state whenever it hits either endpoint. We then construct a marked Poisson process of such killed paths which, when concatenated, produce a pathwise construction of the Wright--Fisher diffusion.