Positivity of Brownian Transition Densities

Let B be a Borel subset of R and let p(t, x, y) be the transition densities of Brownian motion killed on leaving B. Fix x and y in B. If p(t, x, y) is positive for one t, it is positive for every value of t. Some related results are given. 1. Statement of results. Let d ≥ 2, let B be a Borel subset of R, and let p(t, x, y) denote the transition density of d-dimensional Brownian motion killed on exiting B. The purpose of this paper is to examine the question of when p(t, x, y) is positive and when it is 0. If B is an open set, this problem is easy, but we have in mind domains which are similar to the following. Let B(x, r) be the open ball of radius r centered at x and let B = B(0, 1) − ∪i=1B(xi, ri), where {xi} is dense in B(0, 1). In view of the fact that the set {xi} is dense, B has empty interior. However, if the ri are small enough and go to 0 fast enough, there will be infinitely many pairs (x, y) and t > 0 for which p(t, x, y) > 0 (cf. [BB, Sect. 4]). In order for the problem of when p(t, x, y) is positive to make sense, we need to define p(t, x, y) precisely. We let p(t, x, y) = Φ(t, x, y)− E [ Φ(t− τB , XτB , y); τB < t ] , (1.1) 1Research partially supported by a NSERC (Canada) grant 2Research partially supported by NSF grant DMS–9322689