Some fractal properties of Brownian paths (Q1571108)

The present paper reviews some recent developments in the study of properties of Brownian paths, in which the author took part. The subjects are: Average densities: Following work of \textit{T. Bedford} and \textit{A. M. Fisher} [Proc. Lond. Math. Soc., III. Ser. 64, No. 1, 95-124 (1992; Zbl 0706.28009)] one can use summation methods to define local densities of the natural fractal measures associated to one or more Brownian paths. Dimension spectra: Several recent investigations, e.g. by the author and \textit{S. J. Taylor} [Stochastic Processes Appl. 75, No. 2, 249-261 (1998; Zbl 0932.60041)], deal with the problem of finding the dimension of sets of points where the Brownian path exhibits certain extremal or untypical properties. Brownian multifractals: By introducing time changes one can get multifractal occupation measures supported by subsets of the Brownian path. The simplest example is time change by subordination leading to occupation measure of Lévy processes, but the author explains some more involved constructions.