# On unique extension of time changed reflecting Brownian motions - Mathematics > Probability

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 Brownianmotion 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${\bf 1} Dx mxdx$ for a strictly positive continuous integrable function$m$ on $\overline D$. We further show that if there is some $r>0$ so that$D\setminus \overline {B0, r}$ is an unbounded uniform domain, then $Y$admits one and only one symmetric diffusion that genuinely extends it andadmits no killings. In other words, in this case $X$ or equivalently, $Y$ hasa unique Martin boundary point at infinity.

Author: Zhen-Qing Chen, Masatoshi Fukushima

Source: https://arxiv.org/