Open Access
1996 The Dimension of the Frontier of Planar Brownian Motion
Gregory Lawler
Author Affiliations +
Electron. Commun. Probab. 1: 29-47 (1996). DOI: 10.1214/ECP.v1-975


Let $B$ be a two dimensional Brownian motion and let the frontier of $B[0,1]$ be defined as the set of all points in $B[0,1]$ that are in the closure of the unbounded connected component of its complement. We prove that the Hausdorff dimension of the frontier equals $2(1 - \alpha)$ where $\alpha$ is an exponent for Brownian motion called the two-sided disconnection exponent. In particular, using an estimate on $\alpha$ due to Werner, the Hausdorff dimension is greater than $1.015$.


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Gregory Lawler. "The Dimension of the Frontier of Planar Brownian Motion." Electron. Commun. Probab. 1 29 - 47, 1996.


Accepted: 10 March 1996; Published: 1996
First available in Project Euclid: 25 January 2016

zbMATH: 0857.60083
MathSciNet: MR1386292
Digital Object Identifier: 10.1214/ECP.v1-975

Primary: 60J65

Keywords: Brownian motion , frontier , Hausdorff dimension , Random fractals

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