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1996 Hausdorff Dimension of Cut Points for Brownian Motion
Gregory Lawler
Author Affiliations +
Electron. J. Probab. 1: 1-20 (1996). DOI: 10.1214/EJP.v1-2


Let $B$ be a Brownian motion in $R^d$, $d=2,3$. A time $t\in [0,1]$ is called a cut time for $B[0,1]$ if $B[0,t) \cap B(t,1] = \emptyset$. We show that the Hausdorff dimension of the set of cut times equals $1 - \zeta$, where $\zeta = \zeta_d$ is the intersection exponent. The theorem, combined with known estimates on $\zeta_3$, shows that the percolation dimension of Brownian motion (the minimal Hausdorff dimension of a subpath of a Brownian path) is strictly greater than one in $R^3$.


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Gregory Lawler. "Hausdorff Dimension of Cut Points for Brownian Motion." Electron. J. Probab. 1 1 - 20, 1996.


Accepted: 8 November 1995; Published: 1996
First available in Project Euclid: 25 January 2016

zbMATH: 0891.60078
MathSciNet: MR1386294
Digital Object Identifier: 10.1214/EJP.v1-2

Primary: 60J65

Keywords: Brownian motion , cut points , Hausdorff dimension , intersection exponent

Vol.1 • 1996
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