The Library
Hausdorff measure of arcs and Brownian motion on Brownian spatial trees
Tools
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

PDF
WRAP_Croydon_Hausdorff_measure.pdf  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (314Kb) 
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  
Dates: 


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 
Data sourced from Thomson Reuters' Web of Knowledge
Request changes or add full text files to a record
Actions (login required)
View Item 
Downloads
Downloads per month over past year