The Beurling Estimate for a Class of Random Walks
Gregory F Lawler, Cornell University
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 -&epsilon reaches the unit circle without hitting
the curve is bounded above by c &epsilon^{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.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.