Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

Abstract

A Brownian spatial tree is defined to be a pair $(\mathcal{T},ϕ)$, 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 $\mathbb{R}^d$ such that, conditional on $\mathcal{T}$, $ϕ$ 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 $\mathcal{S}:=ϕ(\mathcal{T})$. Applications of this result include the recovery of the spatial tree $(\mathcal{T},ϕ)$ 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.