Potential theory of subordinate killed Brownian motion

TL;DR

This paper develops potential theory for subordinate killed Brownian motions, establishing scale invariant Harnack inequalities and boundary Harnack principles with explicit decay rates under weak scaling conditions.

Contribution

It introduces new boundary Harnack principles for subordinate killed Brownian motions with explicit decay rates, extending potential theory in this context.

Findings

Proves scale invariant Harnack inequality for non-negative harmonic functions.

Establishes two types of boundary Harnack principles with explicit decay rates.

Analyzes boundary and interior behaviors of subordinate killed Brownian motions.

Abstract

Let $W^D$ be a killed Brownian motion in a domain $D\subset {\mathbb R}^d$ and $S$ an independent subordinator with Laplace exponent $\phi$. The process $Y^D$ defined by $Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator $-\phi(-\Delta|_D)$, where $\Delta|_D$ is the Dirichlet Laplacian. In this paper we study the potential theory of $Y^D$ under a weak scaling condition on the derivative of $\phi$. We first show that non-negative harmonic functions of $Y^D$ satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of $Y^D$. The first boundary Harnack principle deals with a $C^{1,1}$ domain $D$ and non-negative functions which are harmonic near the boundary of $D$, while the second one is for a more general domain $D$ and non-negative functions which are harmonic near the boundary of an interior open subset of $D$. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of $Y^D$.