Super-Brownian motion: Path properties and hitting probabilities
type: Item
instance of: scholarly article
title: Super-Brownian motion: Path properties and hitting probabilities
zbMATH Open document ID: 0692.60063
DOI: 10.1007/BF00333147
author: Edwin Perkins
author: Donald A. Dawson
author: Ian Iscoe
publication date: 1989-01-01T00:00:00Z
review text: The model under consideration is a super-Brownian motion \(X=\{X_ t;t\geq 0\}\) in \({\mathbb{R}}^ d\), i.e. a measure-valued diffusion process related to the equation \(\partial u/\partial t=2^{-1}(\Delta u-u^ 2)\) via the Laplace transition functional. The basic goal is the derivation of path properties of \(\{S(X_ t)\); \(t\geq 0\}\) where \(S(X_ t)\) is the closed support of the measure state \(X_ t\) at time t. This includes a modulus of continuity (d\(\geq 1)\), a study of ``polar sets'' and ``multiple points'', and the derivation of an exact Hausdorff measure function for the ``range'' of this support process (d\(\geq 4).\) The super-Brownian motion exhibits a richer family of behaviors than Brownian motion. For instance, points are hit in dimensions \(d\leq 3\), in \(d=4\) multiple points of all orders exist, and for \(d\leq 7\) (and only for these dimensions) there are double points. Proofs are much more complicated as for analogous Brownian motion sample path properties. They include analytic estimates for the hitting of balls by the X process. Also nonstandard analysis arguments are involved.
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 4133258
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: closed support
zbMATH Keywords: multiple points
zbMATH Keywords: polar sets
zbMATH Keywords: measure- valued diffusion process
zbMATH Keywords: Laplace transition functional
zbMATH Keywords: modulus of continuity
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4133258