Transcience/recurrence for normally reflected Brownian motion in   unbounded domains
  Let $D\subsetneq R^d$ be an unbounded domain and let $B(t)$ be a Brownian motion in $D$ with normal reflection at the boundary. We study the transcience/recurrence dichotomy, focusing mainly on domains of the form $D=\{(x,z)\in R^{l+m}:|z|<H(|x|)\}$, where $d=l+m$ and $H$ is a sufficiently regular function. This class of domains includes various horn-shaped domains and generalized slab domains. 
Ross G. Pinsky