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The Exact Hausdorff Measure of Brownian Multiple Points, II

  • Jean-François Le Gall
Chapter
Part of the Progress in Probability book series (PRPR, volume 17)

Abstract

The purpose of this note is to sharpen a result established in [5] concerning the Hausdorff measure of the set of multiple points of a d-dimensiohal Brownian motion. Let X = (Xt, t ≥ 0) denote a standard two-dimensional Brownian motion and, for every integer k ≥ 1, let Mk, denote the set of k-multiple points of X (a point z is said to be k-multiple if there exist k distinct times \(0 \leqslant t_1 < \ldots < t_k \) such that \(X_{t_1 } = \ldots = X_{t_k } = z \)). A canonical measure on Mk, can be constructed as follows. Set: Open image in new window The intersection local time of X with itself, at the order k, is the Radon measure on J k formally defined by:
$${\alpha _{\text{k}}}({\text{d}}{{\text{t}}_1}...{\text{d}}{{\text{t}}_{\text{k}}}) = {\delta _{(0)}}({{\text{X}}_{{{\text{t}}_1}}} - {{\text{X}}_{{{\text{t}}_2}}})...{\delta _{(0)}}({{\text{X}}_{{{\text{t}}_{{\text{k}} - 1}}}} - {{\text{X}}_{{{\text{t}}_{\text{k}}}}})\;{\text{d}}{{\text{t}}_1}...{\text{d}}{{\text{t}}_{\text{k}}} $$
where δ(0) denotes the Dirac measure at 0 in ℝ2. A precise definition of α k may be found in Rosen [7] or Dynkin [2]. As the previous formal definition suggests, the measure α k is supported on the set \(\left\{ {({{\text{t}}_{{1}}},...,{{\text{t}}_{\text{k}}});{{\text{X}}_{{{{\text{t}}_{{1}}}}}} = ... = {{\text{X}}_{{{{\text{t}}_{\text{k}}}}}}} \right\} \) of k-multiple times. Let ℓk denote the image measure of αk by the mapping \(({{\text{t}}_{{1}}},...,{{\text{t}}_{\text{k}}}) \to {{\text{X}}_{{{{\text{t}}_{{1}}}}}} \). It follows that ℓk is supported on Mk. Notice that ℓ is not a Radon measure, but is a countable sum of finite measures.

Keywords

Brownian Motion Radon Measure Hausdorff Measure Borel Subset Dirac Measure 
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References

  1. [1]
    Ciesielski,Z. Taylor,S.J.: 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)MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Dynkin,E.B.: Random fields associated with multiple points of the Brownian motion. J. Funct. Anal. 62, 397–434 (1985)Google Scholar
  3. [3]
    Federer,H.: Geometric Measure Theory. Springer-Verlag, Berlin Heidelberg New-York 1969.MATHGoogle Scholar
  4. [4]
    Le Gall, J.F.: Le comportement du mouvement brownien entre les deux instants où il passe par un point double. J. Funct. Anal. 71, 246–262 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Le Gall,J.F.: The exact Hausdorff measure of Brownian multiple points. Seminar on Stochastic Processes 1986, 107–137. Birkhäuser Boston 1987.Google Scholar
  6. [6]
    Perkins,E.A.: The exact Hausdorff measure of the level sets of Brownian motion. Z. Wahrsch. verw. Gebiete 58, 373–388 (1981)MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Rosen,J.: Self-intersections of random fields. Ann. Probab. 12, 108–119 (1984)MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Taylor,S.J.: The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Camb. Philos. Soc. 60,253–258 (1964)CrossRefMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • Jean-François Le Gall
    • 1
  1. 1.Laboratoire de Probabilités de l’Univ. Paris 6Paris Cedex 05France

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