Open Access
Translator Disclaimer
1998 Percolation Dimension of Brownian Motion in $R^3$
Chad Fargason
Author Affiliations +
Electron. Commun. Probab. 3: 51-63 (1998). DOI: 10.1214/ECP.v3-993

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.

Citation

Download Citation

Chad Fargason. "Percolation Dimension of Brownian Motion in $R^3$." Electron. Commun. Probab. 3 51 - 63, 1998. https://doi.org/10.1214/ECP.v3-993

Information

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

zbMATH: 0907.60069
MathSciNet: MR1641070
Digital Object Identifier: 10.1214/ECP.v3-993

Subjects:
Primary: 60J65

Keywords: boundary dimension , intersection exponent , Percolation dimension

JOURNAL ARTICLE
13 PAGES


SHARE
Back to Top