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August 2017 The spans in Brownian motion
Steven Evans, Jim Pitman, Wenpin Tang
Ann. Inst. H. Poincaré Probab. Statist. 53(3): 1108-1135 (August 2017). DOI: 10.1214/16-AIHP749

## Abstract

For $d\in\{1,2,3\}$, let $(B^{d}_{t};t\geq0)$ be a $d$-dimensional standard Brownian motion. We study the $d$-Brownian span set $\operatorname{Span}(d):=\{t-s;B^{d}_{s}=B^{d}_{t}\mbox{ for some }0\leq s\leq t\}$. We prove that almost surely the random set $\operatorname{Span}(d)$ is $\sigma$-compact and dense in $\mathbb{R}_{+}$. In addition, we show that $\operatorname{Span}(1)=\mathbb{R}_{+}$ almost surely; the Lebesgue measure of $\operatorname{Span}(2)$ is $0$ almost surely and its Hausdorff dimension is $1$ almost surely; and the Hausdorff dimension of $\operatorname{Span}(3)$ is $\frac{1}{2}$ almost surely. We also list a number of conjectures and open problems.

Pour $d\in\{1,2,3\}$, soit $(B_{t}^{d};t\geq0)$ un mouvement brownien standard $d$-dimensionnel. Nous étudions le $d$-ensemble de portée brownienne $\operatorname{Span}(d):=\{t-s;B^{d}_{s}=B^{d}_{t}\mbox{ pour certains }0\leq s\leq t\}$. Nous prouvons que presque sûrement l’ensemble aléatoire $\operatorname{Span}(d)$ est $\sigma$-compact et dense dans $\mathbb{R}_{+}$. De plus, nous montrons que $\operatorname{Span}(1)=\mathbb{R}_{+}$ presque sûrement ; la mesure de Lebesgue de $\operatorname{Span}(2)$ est $0$ presque sûrement et sa dimension de Hausdorff est $1$ presque sûrement ; et la dimension de Hausdorff de $\operatorname{Span}(3)$ est $\frac{1}{2}$ presque sûrement. Nous listons aussi un certain nombre de conjectures et problèmes ouverts.

## Citation

Steven Evans. Jim Pitman. Wenpin Tang. "The spans in Brownian motion." Ann. Inst. H. Poincaré Probab. Statist. 53 (3) 1108 - 1135, August 2017. https://doi.org/10.1214/16-AIHP749

## Information

Received: 15 June 2015; Revised: 19 February 2016; Accepted: 19 February 2016; Published: August 2017
First available in Project Euclid: 21 July 2017

zbMATH: 06775433
MathSciNet: MR3689962
Digital Object Identifier: 10.1214/16-AIHP749

Subjects:
Primary: 28A78, 60J65

Rights: Copyright © 2017 Institut Henri Poincaré

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Vol.53 • No. 3 • August 2017