Traps for reflected Brownian motion

Abstract. Consider an open set D ⊂ Rd, d ≥ 2, and a closed ball B ⊂ D. Let E xTB 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 supx ExTB = ∞. 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. Mathematics Subject Classifications (2000): 60J45, 35P05, 60G17 1