Excursion theory for the Wright-Fisher diffusion

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.