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Mon, 03 Apr 2000 00:00:00 +0200Mon, 03 Apr 2000 00:00:00 +0200Symmetry properties of average densities and tangent measure distributions of measures on the line
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/818
Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finite lower and upper densities the one-sided average densities always agree with one half of the circular average densities at almost every point. We infer this result from a more general formula, which involves the notion of a tangent measure distribution introduced by Bandt and Graf. This formula shows that the tangent measure distributions are Palm distributions and define self-similar random measures in the sense of U. Zähle.Peter Mörterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/818Mon, 03 Apr 2000 00:00:00 +0200Tangent measure distributions of hyperbolic Cantor sets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/819
Tangent measure distributions were introduced by Bandt and Graf as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by contractive mappings, which are not similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models of Bedford and Fisher.Peter Mörters; Daniela Kriegpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/819Mon, 03 Apr 2000 00:00:00 +0200Average densities and linear rectifiability of measures
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/820
We show that a measure in a Euclidean space is linearly rectifiable if and only if the lower 1-density is positive and finite and agrees with the lower average 1-density almost everywhere.Peter Mörterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/820Mon, 03 Apr 2000 00:00:00 +0200Tangent measure distributions of fractal measures
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/821
Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost every point, all s-dimensional tangent measure distributions define statistically self-similar random measures. Consequently, the local geometry of general measures is not different from the local geometry of self-similar sets. We illustrate the strength of this result by showing how it can be used to improve recently proved relations between ordinary and average densities.Peter Mörters; David Preisspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/821Mon, 03 Apr 2000 00:00:00 +0200The average density of planar Brownian motion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/822
We show that the occupation measure on the path of a planar Brownian motion run for an arbitrary finite time intervalhas an average density of order three with respect to thegauge function t^2 log(1/t). This is a surprising resultas it seems to be the first instance where gauge functions other than t^s and average densities of order higher than two appear naturally. We also show that the average densityof order two fails to exist and prove that the density distributions, or lacunarity distributions, of order threeof the occupation measure of a planar Brownian motion are gamma distributions with parameter 2.Peter Mörterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/822Mon, 03 Apr 2000 00:00:00 +0200Density theorems for the intersection local times of planar Brownian motion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/829
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.Peter Mörterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/829Mon, 03 Apr 2000 00:00:00 +0200Pathwise Kallianpur-Robbins laws for Brownian motion in the plane
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/840
The Kallianpur-Robbins law describes the long term asymptotic behaviour of the distribution of the occupation measure of a Brownian motion in the plane. In this paper we show that this behaviour can be seen at every typical Brownian path by choosing either a random time or a random scale according to the logarithmic laws of order three. We also prove a ratio ergodic theorem for small scales outside an exceptional set of vanishing logarithmic density of order three.Peter Mörterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/840Fri, 28 Jan 2000 00:00:00 +0100