The packing measure of the support of super-Brownian motion
type: Item
instance of: scholarly article
title: The packing measure of the support of super-Brownian motion
zbMATH Open document ID: 0848.60078
DOI: 10.1016/0304-4149(95)00033-4
author: Edwin Perkins
author: S. James Taylor
author: J.-F. Le Gall
published in: Stochastic Processes and their Applications
publication date: 1996-10-21T00:00:00Z
review text: It is known that the value of super-Brownian motion in \(R^d\) is given for \(t > 0\) by a multiple of Hausdorff \(\varphi\)-measure restricted to its closed support where \(\varphi (s) = s^2 \log \log {1 \over s}\) for \(d \geq 3\) and \(s^2 \log {1 \over s} \log \log \log {1 \over s}\) for \(d = 2\). The purpose of this paper is to consider the support of super-Brownian motion from the point of view of packing measure. The packing measure [cf. the second author and \textit{C. Tricot}, Real Anal. Exch. 10, 58-67 (1985; Zbl 0578.28007) and Trans. Am. Math. Soc. 288, 679-699 (1985; Zbl 0537.28003)] of a set \(E\) uses small disjoint balls with centres in \(E\) and seeks to maximize the result. The main result is to show that, for \(d \geq 3\) there is no measure function \(\varphi\) such that the \(\varphi\)-packing measure of the closed support is finite and positive. An integral test is derived which distinguishes those \(\varphi\) which make the packing measure 0 or \(\infty\). Partial results are obtained for \(d = 2\). The results are obtained from estimates of lower densities with respect to a packing function using both the canonical representation and Le Gall's path-valued process.
reviewed by: Donald A. Dawson
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 828740
zbMATH Keywords: super-Brownian motion
zbMATH Keywords: packing measure
zbMATH Keywords: canonical representation
zbMATH Keywords: Le Gall's path-valued process
MaRDI profile type: Publication
full work available at URL: 0304-4149(95)00033-4
OpenAlex ID: W2092088267
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description: scientific article; zbMATH DE number 828740