  In this work we adopt a combination of probabilistic approach and analytic
methods to study the fundamental solutions to variations of the Wright-Fisher
equation in one dimension. To be specific, we consider a diffusion equation on
$\left(0,\infty\right)$ whose diffusion coefficient vanishes at the boundary 0,
equipped with the Cauchy initial data and the Dirichlet boundary condition. One
type of diffusion operator that has been extensively studied is the one whose
diffusion coefficient vanishes linearly at 0. Our main goal is to extend the
study to cases when the diffusion coefficient has a general order of
degeneracy. We primarily focus on the fundamental solution to such a degenerate
diffusion equation. In particular, we study the regularity properties of the
fundamental solution near 0, and investigate how the order of degeneracy of the
diffusion operator and the Dirichlet boundary condition jointly affect these
properties. We also provide estimates for the fundamental solution and its
derivatives near 0.
