The Beurling Estimate for a Class of Random Walks
Vlada Limic (University of British Columbia)
Abstract
An estimate of Beurling states that if $K$ is a curve from $0$ to the unit circle in the complex plane, then the probability that a Brownian motion starting at $-\varepsilon$ reaches the unit circle without hitting the curve is bounded above by $c \varepsilon^{1/2}$. This estimate is very useful in analysis of boundary behavior of conformal maps, especially for connected but rough boundaries. The corresponding estimate for simple random walk was first proved by Kesten. In this note we extend this estimate to random walks with zero mean, finite $(3+\delta)$-moment.
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Pages: 846-861
Publication Date: December 13, 2004
DOI: 10.1214/EJP.v9-228
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