Electronic Journal of Probability

Electronic Journal of Probability > Vol. 9 (2004) > Paper 27

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

Pages: 846-861

Published on: December 13, 2004

Bibliography

R. Bass. Probabilistic Techniques in Analysis. Springer-Verlag. (1995) Math. Review 96e:60001
M. Bousquet-M'elou. Walks on the slit plane: other approaches. Adv. Appl. Math. 27 (2001), 242-288. Math. Review 2002j:60076
Y. Fukai. Hitting time of a half line by two-dimensional random walk. Prob. Th. Rel. Fields 128 (2004), 323-346. Math. Review 2004m:60096
Y. Fukai, K. Uchiyama. Potential kernel for two-dimensional random walk. Ann. Probab. 24 (1996), 1979-1992. Math. Review 2004m:60098
H. Kesten. Hitting probabilities of random walks on Z^d. Stoc. Proc. Appl. 25 (1987), 165-184. Math. Review 89a:60163
G. Lawler. Intersections of Random Walks. Birkh"auser. (1991) Math. Review 92f:60122
G. Lawler. A discrete analogue of a theorem of Makarov. Combin. Probab. Comput. 2 (1993), 181-200. Math. Review 95c:31004
G. Lawler, E. Puckette. The intersection exponent for simple random walk. Combin. Probab. Comput. 9 (2000), 441-464. Math. Review 2001m:60102

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Electronic Journal of Probability. ISSN: 1083-6489