The Hausdorff measure of the support of two-dimensional super-Brownian motion
type: Item
instance of: scholarly article
title: The Hausdorff measure of the support of two-dimensional super-Brownian motion
zbMATH Open document ID: 0856.60055
author: Edwin Perkins
author: J.-F. Le Gall
published in: The Annals of Probability
publication date: 1997-01-27T00:00:00Z
review text: Let \(\mathbb{Q}^\gamma_\mu\) denote the law of two-dimensional super-Brownian motion \(Y_t\) \((t\geq 0)\) having branching rate \(\gamma> 0\) and starting from some \(\mu\) in the space \(M_F(\mathbb{R}^2)\) of positive finite Borel measures on \(\mathbb{R}^2\). The main result states that \(Y_t\) can be obtained as deterministic multiple of the \(h\)-Hausdorff measure \(h- m\) restricted to the closed support \(S(Y_t)\) of \(Y_t\), where \[ h(r)= r^2\log^+(1/r) \log^+ \log^+ \log^+(1/r). \] More precisely, there exists a universal constant \(c_0\in (0, \infty)\) such that \[ Y_t(A)= c_0 \gamma h- m(A\cap S(Y_t))\quad \forall\text{ Borel sets }A\subset \mathbb{R}^2,\quad \mathbb{Q}^\gamma_\mu\text{-a.s.}, \] for all \(t> 0\), \(\gamma> 0\) and \(\mu\in M_F(\mathbb{R}^2)\). This theorem is obtained by moment estimates for the excursion measure of Le Gall's Brownian snake process. It complements results of the second author [Ann. Inst. Henri Poincaré, Probab. Stat. 25, No. 2, 205-224 (1989; Zbl 0679.60053)] and \textit{D. Dawson} and the second author [Mem. Am. Math. Soc. 454 (1991; Zbl 0754.60062)] for higher dimensions. There it has been shown that an analogous statement holds on \(\mathbb{R}^d\) with \(d\geq 3\), but with \(h\) replaced by \(\varphi(r)= r^2\log^+ \log^+(1/r)\).
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 897273
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: two-dimensional super-Brownian motion
zbMATH Keywords: Brownian snake process
reviewed by: Alexander Schied
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176987800
OpenAlex ID: W2061018672
DOI: 10.1214/AOP/1176987800
description: scientific article; zbMATH DE number 897273