Dirichlet heat kernel estimates for rectilinear stable processes

TL;DR

This paper investigates the properties and sharp bounds of the transition density functions for rectilinear stable processes in certain domains, providing geometric characterizations and positivity results.

Contribution

It offers a geometric characterization of domains ensuring irreducibility and establishes sharp two-sided bounds for transition densities in $C^{1,1}$ domains.

Findings

Characterization of domains with irreducible process

Strict positivity of transition densities

Sharp two-sided bounds in $C^{1,1}$ domains

Abstract

Let $d \geq 2$, $\alpha \in (0,2)$, and $X$ be the rectilinear $\alpha$-stable process on $\mathbb{R}^d$. We first present a geometric characterization of an open subset $D\subset \mathbb{R}^d$ so that the part process $X^D$ of $X$ in $D$ is irreducible. We then study the properties of the transition density functions of $X^D$, including the strict positivity property as well as their sharp two-sided bounds in $C^{1,1}$ domains in $\mathbb{R}^d$. Our bounds are shown to be sharp for a class of $C^{1,1}$ domains.