Acta Mathematica

Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk

Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni

Full-text: Open access

Note

The first author was partially supported by NSF Grant DMS-9704552-002; the second author was partially supported by NSF Grant DMS-9803597; the third author was supported in part by grants from the NSF and from PSC-CUNY; the work of all authors was supported in part by a US-Israel BSF grant.

Article information

Source
Acta Math., Volume 186, Number 2 (2001), 239-270.

Dates
Received: 10 May 1999
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891402

Digital Object Identifier
doi:10.1007/BF02401841

Mathematical Reviews number (MathSciNet)
MR1846031

Zentralblatt MATH identifier
1008.60063

Rights
2001 © Institut Mittag-Leffler

Citation

Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer. Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk. Acta Math. 186 (2001), no. 2, 239--270. doi:10.1007/BF02401841. https://projecteuclid.org/euclid.acta/1485891402


Export citation

References

  • Bass, R. F. & Griffin, P. S., The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verw. Gebiete, 70 (1985), 417–436.
  • Bingham, N. H., Goldie, C. M. & Teugels, J. L., Regular Variation. Encyclopedia Math. Appl., 27. Cambridge Univ. Press., Cambridge, 1987.
  • Dembo, A., Peres, Y., Rosen, J. & Zeitouni, O., Thick points for spatial Brownian motion: multifractal analysis of occupation measure. Ann. Probab., 28 (2000), 1–35.
  • —, Thin points for Brownian motion. Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 749–774.
  • Dembo, A. & Zeitouni, O., Large Deviations Techniques and Applications, 2nd edition. Appl. Math., 38. Springer-Verlag, New York, 1998.
  • Einmahl, U., Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal., 28 (1989), 20–68.
  • Erdős, P. & Taylor, S. J., Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar., 11 (1960), 137–162.
  • Kahane, J.-P., Some Random Series of Functions, 2nd edition. Cambridge Stud. Adv. Math., 5. Cambridge Univ. Press, Cambridge, 1985.
  • Kaufman, R., Une propriété metriqué du mouvement brownien. C. R. Acad. Sci. Paris Sér. A-B, 268 (1969), A727-A728.
  • Komlós, J., Major, P. & Tusnády, G., An approximation of partial sums of independent RV's, and the sample DF, I. Z. Wahrsch. Verw. Gebiete, 32 (1975), 111–131.
  • Le Gall, J.-F. & Rosen, J., The range of stable random walks. Ann. Probab., 19 (1991), 650–705.
  • Lyons, R. & Pemantle, R., Random walks in a random environment and first-passage percolation on trees. Ann. Probab., 20 (1992), 125–136.
  • Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Stud. Adv. Math., 44. Cambridge Univ. Press, Cambridge, 1995.
  • Mörters, P., The average density of the path of planar Brownian motion. Stochastic Process. Appl., 74 (1998), 133–149.
  • Orey, S. & Taylor, S. J., How often on a Brownian path does the law of the iterated logarithm fail?Proc. London Math. Soc. (3), 28 (1974), 174–192.
  • Pemantle, R., Peres, Y., Pitman, J. & Yor, M., Where did the Brownian particle go? To appear in Electron. J. Probab.
  • Perkins, E. A. & Taylor, S. J., Uniform measure results for the image of subsets under Brownian motion. Probab. Theory Related Fields, 76 (1987), 257–289.
  • Ray, D., Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc., 106 (1963), 436–444.
  • Révész, P., Random Walk in Random and Non-Random Environments. World Sci. Publishing, Teaneck, NJ, 1990.
  • Spitzer, F., Principles of Random Walk. Van Nostrand, Princeton, NJ, 1964.