Electronic Communications in Probability

Percolation Dimension of Brownian Motion in $R^3$

Chad Fargason

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Let $B(t)$ be a Brownian motion in $R^3$. A subpath of the Brownian path $B[0,1]$ is a continuous curve $\gamma(t)$, where $\gamma[0,1] \subseteq B[0,1]$ , $\gamma(0) = B(0)$, and $\gamma(1) = B(1)$. It is well-known that any subset $S$ of a Brownian path must have Hausdorff dimension $\text{dim} (S) \leq 2.$ This paper proves that with probability one there exist subpaths of $B[0,1]$ with Hausdorff dimension strictly less than 2. Thus the percolation dimension of Brownian motion in $R^3$ is strictly less than 2.

Article information

Electron. Commun. Probab., Volume 3 (1998), paper no. 7, 51-63.

Accepted: 27 February 1998
First available in Project Euclid: 2 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Percolation dimension boundary dimension intersection exponent

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Fargason, Chad. Percolation Dimension of Brownian Motion in $R^3$. Electron. Commun. Probab. 3 (1998), paper no. 7, 51--63. doi:10.1214/ECP.v3-993. https://projecteuclid.org/euclid.ecp/1456935913

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