Let $D$ be an unbounded domain in $RR^d$ with $dgeq 3$. 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 $Dsetminus overline {B(0, r)}$ 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.
Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting proce
ss and the value of the drift vector has a product form. Moreover, the first component is the symmetrizing measure on the domain for the reflecting diffusion without inert drift, and the second component has a Gaussian distribution. We also consider processes where the drift is given in terms of the gradient of a potential.
