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The exact packing measure of Brownian double points
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  • Published: 27 November 2007

The exact packing measure of Brownian double points

  • Peter Mörters1 &
  • Narn-Rueih Shieh2 

Probability Theory and Related Fields volume 143, pages 113–136 (2009)Cite this article

  • 116 Accesses

  • 5 Citations

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Abstract

Let \(D\subset {\mathbb{R}}^3\) be the set of double points of a three-dimensional Brownian motion. We show that, if ξ = ξ3(2,2) is the intersection exponent of two packets of two independent Brownian motions, then almost surely, the ϕ-packing measure of D is zero if

$$ \int_{0^+} r^{-1-\xi} \phi(r)^{\xi} \, dr < \infty,$$

and infinity otherwise. As an important step in the proof we show up-to-constants estimates for the tail at zero of Brownian intersection local times in dimensions two and three.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Peter Mörters

  2. Department of Mathematics, National Taiwan University, Taipei, 10617, Taiwan

    Narn-Rueih Shieh

Authors
  1. Peter Mörters
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  2. Narn-Rueih Shieh
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Corresponding author

Correspondence to Peter Mörters.

Additional information

Peter Mörters was supported by Grant EP/C500229/1 and an Advanced Research Fellowship of the EPSRC. Narn-Rueih Shieh was supported by an NSC(Taiwan) grant.

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Cite this article

Mörters, P., Shieh, NR. The exact packing measure of Brownian double points. Probab. Theory Relat. Fields 143, 113–136 (2009). https://doi.org/10.1007/s00440-007-0122-x

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  • Received: 22 September 2005

  • Revised: 10 October 2007

  • Published: 27 November 2007

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0122-x

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Keywords

  • Brownian motion
  • Self-intersections
  • Intersection local time
  • Wiener sausage
  • Lower tail asymptotics
  • Intersection exponent
  • Packing measure
  • Packing gauge

Mathematics Subject Classification (2000)

  • Primary: 60J65
  • 60G17
  • Secondary: 60J55
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