Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

TL;DR

This paper investigates the geometric and probabilistic properties of Brownian spatial trees, establishing an intrinsic metric via Hausdorff measure, and demonstrates how to recover the tree structure and construct Brownian motion on its range, with applications to scaling limits.

Contribution

It introduces a method to define an intrinsic metric on Brownian spatial trees using Hausdorff measure and shows how to recover the tree and Brownian motion from the range, extending to scaling limits.

Findings

Hausdorff measure defines an intrinsic metric in high dimensions.

The spatial tree can be reconstructed from its range alone.

Brownian motion on the tree is the scaling limit of random walks.

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 $\phi$ is a random continuous function from $\mathcal{T}$ into $\mathbb{R}^d$ such that, conditional on $\mathcal{T}$, $\phi$ maps each arc of $\mathcal{T}$ to the image of a Brownian motion path in $\mathbb{R}^d$ 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.