Electronic Communications in Probability

The Dimension of the Frontier of Planar Brownian Motion

Gregory Lawler

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1453756496

Digital Object Identifier
doi:10.1214/ECP.v1-975

Mathematical Reviews number (MathSciNet)
MR1386292

Zentralblatt MATH identifier
0857.60083

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

Keywords
Brownian motion Hausdorff dimension frontier random fractals

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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References

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