Translator Disclaimer
May 2009 Hausdorff measure of arcs and Brownian motion on Brownian spatial trees
David A. Croydon
Ann. Probab. 37(3): 946-978 (May 2009). DOI: 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.

Citation

Download Citation

David A. Croydon. "Hausdorff measure of arcs and Brownian motion on Brownian spatial trees." Ann. Probab. 37 (3) 946 - 978, May 2009. https://doi.org/10.1214/08-AOP425

Information

Published: May 2009
First available in Project Euclid: 19 June 2009

zbMATH: 1219.60052
MathSciNet: MR2537546
Digital Object Identifier: 10.1214/08-AOP425

Subjects:
Primary: 60G57
Secondary: 60J80, 60K35, 60K37

Rights: Copyright © 2009 Institute of Mathematical Statistics

JOURNAL ARTICLE
33 PAGES


SHARE
Vol.37 • No. 3 • May 2009
Back to Top