The spans in Brownian motion

For d ϵ {1, 2, 3}, let (Bdt ; t > 0) be a d-dimensional standard Brownian motion. We study the d-Brownian span set Span(d) := {t - s;Bds = Bdt for some 0 < s < t}. We prove that almost surely the random set Span(d) is α-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 12 almost surely. We also list a number of conjectures and open problems