The spans in Brownian motion

Steven Evans, Jim Pitman, Wenpin Tang
2017 Annales de l'I.H.P. Probabilites et statistiques  
For d ∈{1,2,3}, let (B^d_t; t ≥ 0) be a d-dimensional standard Brownian motion. We study the d-Brownian span set Span(d):={t-s; B^d_s=B^d_t for some 0 ≤ s ≤ t}. We prove that almost surely the random set Span(d) is σ-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.
doi:10.1214/16-aihp749 fatcat:q4bumpqcjjcztjpbx4z7n32u6m