The spans in Brownian motion

作者：Evans Steven^*; Pitman Jim; Tang Wenpin

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2017, 53(3): 1108-1135.

DOI：10.1214/16-AIHP749

• Brownian span set
• Random set
• Energy method
• Fractal projection
• Hausdorff dimension
• Multiple point
• Self-intersection
• Local time
• Self-similar

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.

• 出版日期2017-8