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 Electronic Communications in Probability > Vol. 1 (1996) > Paper 5 open journal systems 


The Dimension of the Frontier of Planar Brownian Motion

Gregory F. Lawler, Duke University


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 $1.015$.


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Pages: 29-47

Published on: March 10, 1996
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Electronic Communications in Probability. ISSN: 1083-589X