The spans in Brownian motion

Abstract

For $d\in \{1,2,3\}$, let $(B_t^d;t\ge 0)$ 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.

33 pages, 4 figures. This paper is published by http://projecteuclid.org/euclid.aihp/1500624032

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Cited by in corpus (1)

• Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion