The fundamental solution to 1D degenerate diffusion equation with one-sided boundary

Abstract

In this work we adopt a combination of probabilistic approach and analytic method to study the fundamental solution to a certain type of one-dimensional degenerate diffusion equation. To be specific, we consider a diffusion equation on (0,∞) whose diffusion coefficient vanishes at the boundary 0, equipped with the Cauchy initial data and the Dirichlet boundary condition. One such diffusion equation 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, with a primary 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.