Electronic Journal of Probability

Hausdorff Dimension of Cut Points for Brownian Motion

Gregory Lawler

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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 $R^3$.

Article information

Electron. J. Probab., Volume 1 (1996), paper no. 2, 20 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]

Brownian motion Hausdorff dimension cut points intersection exponent

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Lawler, Gregory. Hausdorff Dimension of Cut Points for Brownian Motion. Electron. J. Probab. 1 (1996), paper no. 2, 20 pp. doi:10.1214/EJP.v1-2. https://projecteuclid.org/euclid.ejp/1453756465

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