Electronic Journal of Probability

Hausdorff Dimension of Cut Points for Brownian Motion

Gregory Lawler

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1453756465

Digital Object Identifier
doi:10.1214/EJP.v1-2

Mathematical Reviews number (MathSciNet)
MR1386294

Zentralblatt MATH identifier
0891.60078

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

Keywords
Brownian motion Hausdorff dimension cut points intersection exponent

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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