The spans in Brownian motion

For d is an element of {1, 2, 3}, let (B-t(d); t >= 0) be a d-dimensional standard Brownian motion. We study the d-Brownian span set Span(d) := {t - s; B-s(d) = B-t(d). for some 0 <= s <= t}. We prove that almost surely the random set Span(d) is sigma-compact and dense in R+. In addition, we show that Span(1) = R+ almost surely; the Lebesgue measure of Span(2) is 0 almost surely and its Hausdorff dimension is 1 almost surely; and the Hausdorff dimension of Span(3) is 1/2 almost surely. We also list a number of conjectures and open problems.