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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


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.


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


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

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

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

Keywords: Branching random walk , Dawson–Watanabe super-process , diffusion , Hausdorff measure , random environment , Random walk , Scaling limit , Spatial tree

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 3 • May 2009
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