The Fundamental Solution to One-Dimensional Degenerate Diffusion Equation, I

TL;DR

This paper investigates the fundamental solutions of one-dimensional degenerate diffusion equations, extending existing models to cases with general degeneracy orders, and analyzes their regularity and boundary behavior using probabilistic and analytic methods.

Contribution

It introduces a comprehensive analysis of fundamental solutions for degenerate diffusion equations with general degeneracy, expanding beyond linear cases and exploring boundary regularity and estimates.

Findings

Derived regularity properties of fundamental solutions near boundary 0

Provided estimates for solutions and derivatives near degeneracy point

Extended understanding of boundary effects in degenerate diffusion equations

Abstract

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.