The spans in Brownian motion
نویسندگان
چکیده
For d ∈ {1,2,3}, let (B t ; t ≥ 0) be a d-dimensional standard Brownian motion. We study the d-Brownian span set Span(d) := {t − s;Bd s = B 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 2 almost surely. We also list a number of conjectures and open problems. Résumé. Pour d ∈ {1,2,3}, soit (B t ; t ≥ 0) un mouvement brownien standard d-dimensionnel. Nous étudions le d-ensemble de portée brownienne Span(d) := {t − s;Bd s = B t pour certains 0 ≤ s ≤ t}. Nous prouvons que presque sûrement l’ensemble aléatoire Span(d) est σ -compact et dense dans R+. De plus, nous montrons que Span(1)= R+ presque sûrement ; la mesure de Lebesgue de Span(2) est 0 presque sûrement et sa dimension de Hausdorff est 1 presque sûrement ; et la dimension de Hausdorff de Span(3) est 2 presque sûrement. Nous listons aussi un certain nombre de conjectures et problèmes ouverts. MSC: 28A78; 60J65
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