Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The spans in Brownian motion

Steven Evans, Jim Pitman, and Wenpin Tang

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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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1108-1135.

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624032

Digital Object Identifier
doi:10.1214/16-AIHP749

Mathematical Reviews number (MathSciNet)
MR3689962

Zentralblatt MATH identifier
06775433

Subjects
Primary: 28A78: Hausdorff and packing measures 60J65: Brownian motion [See also 58J65]

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

Citation

Evans, Steven; Pitman, Jim; Tang, Wenpin. The spans in Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1108--1135. doi:10.1214/16-AIHP749. https://projecteuclid.org/euclid.aihp/1500624032


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