Some fractal properties of Brownian paths
type: Item
instance of: scholarly article
title: Some fractal properties of Brownian paths
zbMATH Open document ID: 0958.60036
author: Narn-Rueih Shieh
published in: Taiwanese Journal of Mathematics
publication date: 2001-03-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 1472435
zbMATH Keywords: Brownian motion
zbMATH Keywords: Brownian path
zbMATH Keywords: intersection of Brownian paths
zbMATH Keywords: level set
zbMATH Keywords: Brownian multifractal
zbMATH Keywords: dilation-stable Lévy process
zbMATH Keywords: process with stable component
zbMATH Keywords: average density
zbMATH Keywords: dimension spectrum
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407197
OpenAlex ID: W1584144960
DOI: 10.11650/TWJM/1500407197
description: scientific article; zbMATH DE number 1472435