The spans in Brownian motion

TL;DR

This paper investigates the properties of the span sets of Brownian motion in dimensions 1, 2, and 3, revealing their measure, density, and fractal dimensions, and establishing almost sure behaviors.

Contribution

It provides a detailed analysis of the geometric and measure-theoretic properties of Brownian span sets across different dimensions, including new results on their Hausdorff dimensions.

Findings

Span(1) equals all positive real numbers almost surely

Lebesgue measure of Span(2) is zero almost surely

Hausdorff dimension of Span(3) is 1/2 almost surely

Abstract

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