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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. doi:10.1214/08-AOP425 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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | May 2009 | ||||
Dates: |
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Volume: | Vol.37 | ||||
Number: | No.3 | ||||
Page Range: | pp. 946-978 | ||||
DOI: | 10.1214/08-AOP425 | ||||
Status: | Peer Reviewed | ||||
Access rights to Published version: | Restricted or Subscription Access |
Data sourced from Thomson Reuters' Web of Knowledge
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