Traps for reflected Brownian motion

Consider an open set [InlineMediaObject not available: see fulltext.], d ≥ 2, and a closed ball [InlineMediaObject not available: see fulltext.]. Let [InlineMediaObject not available: see fulltext.] denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x D. We say that D is a trap domain if [InlineMediaObject not available: see fulltext.]. A domain D is not a trap domain if and only if the reflecting Brownian motion in D is uniformly ergodic. We fully characterize the simply connected planar trap domains using a geometric condition and give a number of (less complete) results for d > 2.