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On the most visited sites of planar Brownian motion


 
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1. Title Title of document On the most visited sites of planar Brownian motion
 
2. Creator Author's name, affiliation, country Valentina Cammarota; "Sapienza" University of Rome; Italy
 
2. Creator Author's name, affiliation, country Peter Mörters; University of Bath; United Kingdom
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Brownian motion, Hausdorff dimension, Hausdorff gauge, local time, point of infinite multiplicity, uniform dimension estimates.
 
3. Subject Subject classification 60J65
 
4. Description 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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7. Date (YYYY-MM-DD) 2012-04-10
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1809
 
10. Identifier Digital Object Identifier 10.1214/ECP.v17-1809
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 17
 
12. Language English=en en
 
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