## 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.

Verfasserangaben: Peter Mörters urn:nbn:de:hbz:386-kluedo-7988 Preprints (rote Reihe) des Fachbereich Mathematik (303) Preprint Englisch 1999 1999 Technische Universität Kaiserslautern 03.04.2000 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 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Axx Classical measure theory / 28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Axx Classical measure theory / 28A80 Fractals [See also 37Fxx] 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G17 Sample path properties 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Jxx Markov processes / 60J65 Brownian motion [See also 58J65] Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011
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