Electronic Communications in Probability

Where does a random process hit a fractal barrier?

Itai Benjamini and Alexander Shamov

Full-text: Open access

Abstract

Given a Brownian path $\beta (t)$ on $\mathbb{R} $, starting at $1$, a.s. there is a singular time set $T_{\beta }$, such that the first hitting time of $\beta $ by an independent Brownian motion, starting at $0$, is in $T_{\beta }$ with probability one. A couple of problems regarding hitting measure for random processes are presented.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 25, 5 pp.

Dates
Received: 4 August 2016
Accepted: 5 April 2018
First available in Project Euclid: 28 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1524881133

Digital Object Identifier
doi:10.1214/18-ECP131

Mathematical Reviews number (MathSciNet)
MR3798236

Zentralblatt MATH identifier
1398.60087

Subjects
Primary: subversive math

Keywords
fractal harmonic measure

Rights
Creative Commons Attribution 4.0 International License.

Citation

Benjamini, Itai; Shamov, Alexander. Where does a random process hit a fractal barrier?. Electron. Commun. Probab. 23 (2018), paper no. 25, 5 pp. doi:10.1214/18-ECP131. https://projecteuclid.org/euclid.ecp/1524881133


Export citation

References

  • [1] I. Benjamini and A. Yadin. Harmonic measure in the presence of a spectral gap. Annales Institut Henri Poincare. 52, 1050–1060, 2016.
  • [2] J. Bourgain. On the Hausdorff dimension of harmonic measure in higher dimension. Invent. Math. 87, 477–483, 1987.
  • [3] K. Falconer and J. Fraser, The visible part of plane self-similar sets. Proc. Amer. Math. Soc. 141, 269–278, 2013.
  • [4] J. Garnett and D. Marshall. Harmonic measure, volume 2. Cambridge University Press, 2005.
  • [5] P. Jones and T. Wolff. Hausdorff dimension of harmonic measures in the plane. Acta Mathematica. 161, 131–144, 1988.
  • [6] G. Lawler. A discrete analogue of a theorem of Makarov. Combin. Probab. Comput. 2, 181–199, 1993
  • [7] N. Makarov. On the distortion of boundary sets under conformal mappings. Proc. London Math. Soc. 3, 369–384, 1985.
  • [8] D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer, 1999