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2012 On the most visited sites of planar Brownian motion
Valentina Cammarota, Peter Mörters
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Electron. Commun. Probab. 17: 1-9 (2012). DOI: 10.1214/ECP.v17-1809

Abstract

Let $(B_t \colon t \ge 0)$ be a planar Brownian motion and define gauge functions $\phi_\alpha(s)=\log(1/s)^{-\alpha}$ for $\alpha>0$. If $\alpha<1$ we show that almost surely there exists a point $x$ in the plane such that ${\mathcal H}^{\phi_\alpha}(\{t \ge 0 \colon B_t=x\})>0$,but if $\alpha>1$ almost surely ${\mathcal H}^{\phi_\alpha} (\{t \ge 0 \colon B_t=x\})=0$ simultaneously for all $x\in{\mathbb R}^2$. This resolves a longstanding open problem posed by S.J. Taylor in 1986.

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Valentina Cammarota. Peter Mörters. "On the most visited sites of planar Brownian motion." Electron. Commun. Probab. 17 1 - 9, 2012. https://doi.org/10.1214/ECP.v17-1809

Information

Accepted: 10 April 2012; Published: 2012
First available in Project Euclid: 7 June 2016

zbMATH: 1244.60079
MathSciNet: MR2910986
Digital Object Identifier: 10.1214/ECP.v17-1809

Subjects:
Primary: 60J65

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