Traps for reflected Brownian motion

Consider an open set [inline-graphic not available: see fulltext], d ≥ 2, and a closed ball [inline-graphic not available: see fulltext]. Let [inline-graphic 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 [inline-graphic 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.