Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

David A. Croydon

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

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 ℝ^d such that, conditional on $\mathcal{T}$ , φ maps each arc of $\mathcal{T}$ to the image of a Brownian motion path in ℝ^d 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.

Primary Subjects: 60G57

Secondary Subjects: 60J80, 60K35, 60K37

Keywords: Spatial tree; Dawson–Watanabe super-process; Hausdorff measure; diffusion; random environment; random walk; scaling limit; branching random walk

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1245434025
Digital Object Identifier: doi:10.1214/08-AOP425
Zentralblatt MATH identifier: 05587820
Mathematical Reviews number (MathSciNet): MR2537546

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