On unique extension of time changed reflecting Brownian motions

TL;DR

This paper investigates the conditions under which reflecting Brownian motions on unbounded domains in orm{R}^d are transient and establishes the uniqueness of their symmetric diffusion extensions under certain geometric conditions.

Contribution

It proves that RBM on domains containing an unbounded uniform domain is transient and establishes the uniqueness of symmetric diffusion extensions for certain time-changed RBMs.

Findings

RBM on domains with unbounded uniform subdomains is transient.

Under specific conditions, the time-changed RBM admits a unique symmetric diffusion extension.

The extension has a single Martin boundary point at infinity.

Abstract

Let $D$ be an unbounded domain in $\RR^d$ with $d\geq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $\overline D$ is transient. Next assume that RBM $X$ on $\overline D$ is transient and let $Y$ be its time change by Revuz measure ${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $\overline D$. We further show that if there is some $r>0$ so that $D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity.