Density theorems for the intersection local times of planar Brownian motion

  • We show that the intersection local times \(\mu_p\) on the intersection of \(p\) independent planar Brownian paths have an average density of order three with respect to the gauge function \(r^2\pi\cdot (log(1/r)/\pi)^p\), more precisely, almost surely, \[ \lim\limits_{\varepsilon\downarrow 0} \frac{1}{log |log\ \varepsilon|} \int_\varepsilon^{1/e} \frac{\mu_p(B(x,r))}{r^2\pi\cdot (log(1/r)/\pi)^p} \frac{dr}{r\ log (1/r)} = 2^p \mbox{ at $\mu_p$-almost every $x$.} \] We also show that the lacunarity distributions of \(\mu_p\), at \(\mu_p\)-almost every point, is given as the distribution of the product of \(p\) independent gamma(2)-distributed random variables. The main tools of the proof are a Palm distribution associated with the intersection local time and an approximation theorem of Le Gall.

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Author:Peter Mörters
URN (permanent link):urn:nbn:de:hbz:386-kluedo-7988
Serie (Series number):Preprints (rote Reihe) des Fachbereich Mathematik (303)
Document Type:Preprint
Language of publication:English
Year of Completion:1999
Year of Publication:1999
Publishing Institute:Technische Universität Kaiserslautern
Tag:Brownian motion ; Palm distribution ; average density ; density distribution ; intersection local time ; lacunarity distribution ; logarithmic average
The paper is a continuation of Number 296 of the same series and will be embedded in a larger joint project with N.R.Shieh (Taipeh).
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:510 Mathematik
MSC-Classification (mathematics):28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
28A80 Fractals [See also 37Fxx]
60G17 Sample path properties
60J65 Brownian motion [See also 58J65]

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