## Electronic Communications in Probability

### Percolation Dimension of Brownian Motion in $R^3$

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ecp/1456935913

Digital Object Identifier
doi:10.1214/ECP.v3-993

Mathematical Reviews number (MathSciNet)
MR1641070

Zentralblatt MATH identifier
0907.60069

Subjects

Rights

#### Citation

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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