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Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

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Croydon, David A.. (2009) Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. Annals of Probability, Vol.37 (No.3). pp. 946-978. ISSN 0091-1798

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Official URL: http://dx.doi.org/10.1214/08-AOP425

Abstract

A Brownian spatial tree is defined to be a pair $(\mathcal{T},\phi)$, where $\mathcal{T}$ is the rooted real tree naturally associated with a Brownian excursion and φ is a random continuous function from $\mathcal{T}$ into ℝd such that, conditional on $\mathcal{T}$, φ maps each arc of $\mathcal{T}$ to the image of a Brownian motion path in ℝd run for a time equal to the arc length. It is shown that, in high dimensions, the Hausdorff measure of arcs can be used to define an intrinsic metric $d_{\mathcal{S}}$ on the set $\mathcal{S}:=\phi(\mathcal{T})$. Applications of this result include the recovery of the spatial tree $(\mathcal{T},\phi)$ from the set $\mathcal{S}$ alone, which implies in turn that a Dawson–Watanabe super-process can be recovered from its range. Furthermore, $d_{\mathcal{S}}$ can be used to construct a Brownian motion on $\mathcal{S}$, which is proved to be the scaling limit of simple random walks on related discrete structures. In particular, a limiting result for the simple random walk on the branching random walk is obtained.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Stochastic processes, Hausdorff measures, Measure theory, Random walks (Mathematics), Scaling (Social sciences)
Journal or Publication Title:	Annals of Probability
Publisher:	Institute of Mathematical Statistics
ISSN:	0091-1798
Date:	May 2009
Volume:	Vol.37
Number:	No.3
Page Range:	pp. 946-978
Identification Number:	10.1214/08-AOP425
Status:	Peer Reviewed
Access rights to Published version:	Restricted or Subscription Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/2260