On unique extension of time changed reflecting Brownian motions

Abstract

Let $D$ be an unbounded domain in $\RR^d$ with $d\ge 3$. 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 $1_D(x)m(x)dx$ 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.

arXiv commentTo appear in Ann. Inst. Henri Poincare Probab. Statist