Advertisement

The Exact Hausdorff Measure of Brownian Multiple Points

  • Jean-François Le Gall
Part of the Progress in Probability and Statistics book series (PRPR, volume 13)

Abstract

Let B = (Bt,t ≧0) denote a d-dimensional Brownian motion, with d ≧ 2. Dvoretzky, Erdös, Kakutani and Taylor [2,3,4] have proved that the path of B has points of multiplicity p ≧ 2 if and only if one of the two following conditions is satisfied:
$$\frac{-d=2,p arbitrary}{-d=3,p=2} $$
(1.a)

Keywords

Brownian Motion Sample Path Multiple Point Hausdorff Measure Borel Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. CIESIELSKI and S.J. TAYLOR. First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103, 434–450 (1962).MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. DVORETSKY, P. ERDOS and S. KAKUTANI. Double points of paths of Brownian motion in n-.space. Acta Sci. Math. (Szeged) 12, 64–81 (1950).Google Scholar
  3. [3]
    A. DVORETBKY, P. ERDÖS and S. KAKUTANI. Multiple points of paths of Brownian motion in the plane. Bull. Res. Council Isr. Sect. F 3, 364–371 (1954).Google Scholar
  4. [4]
    A. DVORETSKY, P. ERDÖS, S. KAKUT’ANI and S. J. TAYLOR. Triple points of Brownian motion in 3-space. Proc. Carob. Philos. Soc. 53, 856–862 (1957).CrossRefGoogle Scholar
  5. [5]
    E.B. DYNKIN. Additive functionals of several time-reversible Markov processes. J. Funct. Anal. 42, 64–101 (1981).MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    E.B. DYNKIN. Random fields associated with multiple points of the Brownian motion. J. Funct. Anal. 62, 397–434 (1985).MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    B. FRISTEDT. An extension of a theorem of S.J. Taylor concerning the multiple points of the symmetric stable process. 2. Wahrsch. verw. Gebiete 9, 62–64 (1967).MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    D. GEMAN, J. HOROWIT2 and J. ROSEN. A local time analysis of intersections of Brownian paths in the plane. Ann. Probab. 12, 86–107, (1984).MathSciNetMATHGoogle Scholar
  9. [9]
    J.F. LE GALL. Sur la saucisse de Wiener et les points multiples du mouvement brownien. Preprint (1984), to appear in Ann. Probab.Google Scholar
  10. [10]
    J.F. LE GALL. Sur la mesure de Hausdorff de la courbe brownienne. In: Séminaire de Probabilités XIX. Lect. Notes Math. 1123, p. 297–313. Berlin, Heidelberg, New-York: Springer 1985.CrossRefGoogle Scholar
  11. [11]
    J.F. LE GALL. Sur le temps local d’intersection du mouvement • brownien plan et la méthode de renormalisation de Varadhan. In: Séminaire de Probabilités XIX. Lect. Notes Math. 1123, p. 314–331. Berlin, Heidelberg, New-York, Springer 1985.CrossRefGoogle Scholar
  12. [12]
    J.F. LE GALL. Propriétés d’intersection des marches aléatoires, I. Convergence vers le temps local d’intersection. Comm. Math. Phys. 104, 471–507 (1986).MATHGoogle Scholar
  13. [13]
    J.F. LE GALL. Le comportement du mouvement brownien entre les deux instants où il passe par un point double. Preprint (1985), to appear in J. Funct. Anal.Google Scholar
  14. [14]
    P. LEVY. La mesure de Hausdorff de la courbe du mouvement brownien. Giorn. Ist. Ital. Attuari, 16, 1–37 (1953).MathSciNetGoogle Scholar
  15. [15]
    D. RAY. Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc. 106, 436–444 (1963).MATHGoogle Scholar
  16. [16]
    C.A. ROGERS, S.J. TAYLOR. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8, 1–31 (1961).MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    J. ROSEN. Self-intersections of random fields. Ann. Probab. 12, 108–119 (1984).MATHGoogle Scholar
  18. [18]
    J. ROSEN. A local time approach to the self-intersections of Brownian paths in space. Comm. Math. Phys. 88, 327–338 (1983).MATHGoogle Scholar
  19. [19]
    S.J. TAYLOR. The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Carob. Philos. Soc. 60, 253–258 (1964).MATHCrossRefGoogle Scholar
  20. [20]
    S.J. TAYLOR. Multiple points for the sample paths of the symmetric stable process. Z. Wahrsch. Verw. Gebiete 5, 247–264 (1966).CrossRefGoogle Scholar
  21. [21]
    S.J. TAYLOR. Sample path properties of processes with stationary independent increments. In: Stochastic Analysis, Kendall, D. and Harding, E. (eds). London, J. Wiley and Sons 1973.Google Scholar

Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • Jean-François Le Gall
    • 1
  1. 1.Laboratoire de ProbabilitésUniversite P. et M. CurieParis Cedex 05France

Personalised recommendations