Traps for Reflected Brownian Motion

Abstract

Consider an open set D [is an element of the set] R [The set of Real Numbers] [superscript]d, d [is greater than or equal to] 2, and a closed ball B [is a proper subset of] D. Let E[superscript]xT[subscript]B denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x [is an element of the set] D. We say that D is a trap domain if sup[subscript]x E[superscript]xT[subscript]B = [infinity]. We fully characterize simply connected planar trap domains using a geometric condition. We give a number of (less complete) results for multidimensional domains. We discuss the relationship between trap domains and some other potential theoretic properties of D such as compactness of the 1-resolvent of the Neumann Laplacian. In addition, we give an answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity for the Dirichlet Laplacian in a domain D and compactness of the 1-resolvent of the Neumann Laplacian in D.