## Annals of Probability

### Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

David A. Croydon

#### 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.

#### Article information

Source
Ann. Probab., Volume 37, Number 3 (2009), 946-978.

Dates
First available in Project Euclid: 19 June 2009

https://projecteuclid.org/euclid.aop/1245434025

Digital Object Identifier
doi:10.1214/08-AOP425

Mathematical Reviews number (MathSciNet)
MR2537546

Zentralblatt MATH identifier
1219.60052

#### Citation

Croydon, David A. Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. Ann. Probab. 37 (2009), no. 3, 946--978. doi:10.1214/08-AOP425. https://projecteuclid.org/euclid.aop/1245434025

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