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.

Author: Peter Mörters urn:nbn:de:hbz:386-kluedo-7988 Preprints (rote Reihe) des Fachbereich Mathematik (303) Preprint English 1999 1999 Technische Universität Kaiserslautern 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). Fachbereich Mathematik 510 Mathematik 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]
$Rev: 12793$