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The Packing Measure of Planar Brownian Motion

  • J.-F. Le Gall
  • S. James Taylor
Part of the Progress in Probability and Statistics book series (PRPR, volume 13)

Abstract

In a recent paper [4] a new fractal measure φ-p(E) with respect to the monotone function φ(s) was defined,1and it was shown that φ1(s) = s2[log log 1/s]−1 is the right function for measuring Bd(t), the Brownian motion in ℝd, d ≥ 3, in the sense that there are finite positive constants cd such that
$${{\varphi }_{1}}-p\ {{B}^{d}}(A)={{c}_{d}}\left| A \right|a.s.$$
(1)
for every Borel A ⊂ ℝ+ = [0,∞), where |A| denotes the Lebesgue measure of A. This new measure φ-p(E), which we called P-packing measure, involves maximising the φ content of a packing by disjoint balls centered in E, with radii at most δ, and taking the limit as δ ↓ 0. In the present note we consider the problem of determining a function φ2 which gives the analogue of (1) for the critical case of planar Brownian motion B2(t), which we will henceforth denote by Z(t). In fact, we show that no such φ2 exists, and give a test which determines whether φ-p Z[0,1] is 0 or + ∞. This provides a complete solution to Problem 1 of [5].

Keywords

Brownian Motion Sample Path Packing Measure Occupation Measure Brownian Path 
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.

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References

  1. 1.
    A. Dvoretzky and P. Erdös. Some problems on random walk in space. Proc. Second Berkeley Symp. Berkeley (1950), 353–267.Google Scholar
  2. 2.
    J-F. Le Gall. Sur la mesure de Hausdorff de la courbe Brownienne. Séminaire de Probabilités XIX. Springer Lecture Notes 1123 (1985), 297–313.Google Scholar
  3. 3.
    D. Ray. Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc. 106 (1963), 436–444.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    S. J. Taylor and C. Tricot. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. J. Taylor. The use of packing measure in the analysis of random sets. Proc. of 15th Symposium on Stochastic Processes (1985)Google Scholar

Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • J.-F. Le Gall
    • 1
    • 2
  • S. James Taylor
    • 1
    • 2
  1. 1.Laboratoire de ProbabilitésUniversité de Paris VIParisFrance
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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