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. As an application of our results we determine the Hausdorff dimension of the set of times spent at a given boundary point. This result is then used to provide a new proof that Wright–Fisher path laws with different mutation parameters are mutually singular.