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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. 946978. ISSN 00911798

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Official URL: http://dx.doi.org/10.1214/08AOP425
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 superprocess 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:  00911798 
Official Date:  May 2009 
Volume:  Vol.37 
Number:  No.3 
Page Range:  pp. 946978 
Identification Number:  10.1214/08AOP425 
Status:  Peer Reviewed 
Access rights to Published version:  Restricted or Subscription Access 
References:  [1] D. Aldous, The continuum random tree. I, Ann. Probab. 19 (1991), no. 1, 1{28. 
URI:  http://wrap.warwick.ac.uk/id/eprint/2260 
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