Electronic Communications in Probability

The Dimension of the Frontier of Planar Brownian Motion

Gregory Lawler

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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 $1.015$.

Article information

Electron. Commun. Probab., Volume 1 (1996), paper no. 5, 29-47.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Brownian motion Hausdorff dimension frontier random fractals

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Lawler, Gregory. The Dimension of the Frontier of Planar Brownian Motion. Electron. Commun. Probab. 1 (1996), paper no. 5, 29--47. doi:10.1214/ECP.v1-975. https://projecteuclid.org/euclid.ecp/1453756496

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