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