Fundamental Solution to 1D Degenerate Diffusion Equation with Locally Bounded Coefficients

TL;DR

This paper constructs and analyzes the fundamental solution for a class of 1D degenerate diffusion equations with locally bounded coefficients, extending existing results to more general degeneracy orders using probabilistic and analytic methods.

Contribution

It provides an explicit construction and properties of the fundamental solution for degenerate diffusion equations with order b1c2, generalizing previous work mainly focused on b1=1.

Findings

Explicit fundamental solution construction for b1c2

Approximation of the heat kernel near zero with error estimates

Extension of well-posedness results for stochastic differential equations

Abstract

In this work we study the degenerate diffusion equation $\partial_{t}=x^{\alpha}a\left(x\right)\partial_{x}^{2}+b\left(x\right)\partial_{x}$ for $\left(x,t\right)\in\left(0,\infty\right)^{2}$, equipped with a Cauchy initial data and the Dirichlet boundary condition at $0$. We assume that the order of degeneracy at 0 of the diffusion operator is $\alpha\in\left(0,2\right)$, and both $a\left(x\right)$ and $b\left(x\right)$ are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution $p\left(x,y,t\right)$ and prove several properties for $p\left(x,y,t\right)$; by conducting a localization procedure, we obtain an approximation for $p\left(x,y,t\right)$ for $x,y$ in a neighborhood of 0 and $t$ sufficiently small, where the error estimates only rely on the local bounds of $a\left(x\right)$ and $b\left(x\right)$ (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of $\alpha=1$. Our work extends part of the existing results to cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probability view (e.g., wellposedness of stochastic differential equations).