The Beurling estimate for a class of random walks

2004/03/18 by Gregory F. Lawler, Lawler, Gregory F., Vlada Limic +1 · 1 citation
Decision Sciences · Mathematics · #60F99 #60G50 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR #msc:60F99 #msc:60G50

paper · pdf · doi:10.48550/arxiv.math/0403309

17 pages

arxiv created 2004/03/18 · openalex publication_date 2004/03/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 -eps reaches the unit circle without hitting the curve is bounded above by c eps^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, and finite (3+delta) moment.

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