## The average density of planar Brownian motion

• 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.

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Author: Peter Mörters urn:nbn:de:hbz:386-kluedo-7912 Preprints (rote Reihe) des Fachbereich Mathematik (296) Preprint English 1997 1997 Technische Universität Kaiserslautern Brownian motion ; average density ; density distribution ; lacunarity distribution; logarithmic averages ; occupation measure ; order-three density 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]

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