The Annals of Probability

What is the probability of intersecting the set of Brownian double points?

Robin Pemantle and Yuval Peres

Full-text: Open access

Abstract

We give potential theoretic estimates for the probability that a set A contains a double point of planar Brownian motion run for unit time. Unlike the probability for A to intersect the range of a Markov process, this cannot be estimated by a capacity of the set A. Instead, we introduce the notion of a capacity with respect to two gauge functions simultaneously. We also give a polar decomposition of A into a set that never intersects the set of Brownian double points and a set for which intersection with the set of Brownian double points is the same as intersection with the Brownian path.

Article information

Source
Ann. Probab., Volume 35, Number 6 (2007), 2044-2062.

Dates
First available in Project Euclid: 8 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1191860415

Digital Object Identifier
doi:10.1214/009117907000000169

Mathematical Reviews number (MathSciNet)
MR2353382

Zentralblatt MATH identifier
1131.60071

Subjects
Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Keywords
Capacity polar decomposition multiparameter Brownian motion regular point

Citation

Pemantle, Robin; Peres, Yuval. What is the probability of intersecting the set of Brownian double points?. Ann. Probab. 35 (2007), no. 6, 2044--2062. doi:10.1214/009117907000000169. https://projecteuclid.org/euclid.aop/1191860415


Export citation

References

  • Benjamini, I., Pemantle, R. and Peres, Y. (1995). Martin capacity for Markov chains. Ann. Probab. 23 1332--1346.
  • Carleson, L. (1967). Selected Problems on Exceptional Sets. Van Nostrand, Princeton--Toronto--London.
  • Fitzsimmons, P. and Salisbury, T. (1989). Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 25 325--350.
  • Pemantle, R. and Peres, Y. (1995). Galton--Watson trees with the same mean have the same polar sets. Ann. Probab. 23 1102--1124.
  • Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177 417--434.
  • Salisbury, T. (1996). Energy, and intersections of Markov chains. In Random Discrete Structures 213--225. IMA Vol. Math. Appl. 76. Springer, New York.