Electronic Journal of Probability

The Beurling Estimate for a Class of Random Walks

Gregory Lawler and Vlada Limic

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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.

Article information

Electron. J. Probab., Volume 9 (2004), paper no. 27, 846-861.

Accepted: 13 October 2004
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F99: None of the above, but in this section

Beurling projection random walk Green's function escape probabilities

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Lawler, Gregory; Limic, Vlada. The Beurling Estimate for a Class of Random Walks. Electron. J. Probab. 9 (2004), paper no. 27, 846--861. doi:10.1214/EJP.v9-228. https://projecteuclid.org/euclid.ejp/1465229713

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