Dirichlet heat kernel estimates for rectilinear stable processes

Abstract

Let $d\ge 2$, $\alpha \in (0,2)$, and X be the rectilinear α-stable process on $R^d$. We first present a geometric characterization of open subset $D\subset R^d$ 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 $R^d$. Our bounds are shown to be sharp for a class of $C^{1,1}$ domains.