Dirichlet heat kernel estimates for rectilinear stable processes

TL;DR

This paper analyzes Dirichlet heat kernels for the rectilinear α-stable process, establishing a geometric irreducibility criterion for the killed process $X^D$ and proving joint Hölder regularity of the Dirichlet heat kernel $p_D(t,x,y)$. It derives sharp two-sided bounds in $C^{1,1}$ domains under an additional geometric condition, and shows that irreducibility and two-sided estimates fail in general without such geometry, supported by carefully constructed examples. The authors combine probabilistic Lévy-system techniques, exit-time arguments, and analytic PDE methods to obtain upper and lower Dirichlet heat kernel bounds, along with large-time asymptotics via the first eigenvalue $\lambda_1(D)$ for bounded domains. Overall, the work reveals how the anisotropic, coordinate-axis-jump structure of the rectilinear stable process shapes boundary behavior, connectivity, and heat-kernel estimates, highlighting the necessity of domain-geometry assumptions for sharp results. The results have implications for potential theory and boundary regularity for non-local, non-rotationally invariant jump processes.

Abstract

Let $d \geq 2$, $α\in (0,2)$, and $X$ be the rectilinear $α$-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.

Figures (9)

• Figure 1: The set $D := B(X_0,r) \cup B(Y_0,r)$ with $r = 1$, and the set $D:=$ the cubes with round corners in $\mathbb{R}^2$
• Figure 2: The set $D$ is the union of four squares with round corners in $\mathbb{R}^2$
• Figure 3: The points $Q$, $x$ and $x_0$, etc
• Figure 4: The points $Q$ and $x$, and the set $D_Q(\delta_0,r_0)$, etc
• Figure 5: The case when $|y^{(i_k)} - x^{(i_k)}| \le a_5 t^{1/\alpha}$

Theorems & Definitions (59)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• proof