Hausdorff measure of arcs and Brownian motion on brownian spatial trees

Abstract

A Brownian spatial tree is defined to be a pair (T ,φ), where T is the rooted real tree naturally associated with a Brownian excursion and φ is a random continuous function from T into ℝd such that, conditional on T , φ maps each arc of 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 dS on the set S := φ(T ). Applications of this result include the recovery of the spatial tree (T ,φ) from the set S alone, which implies in turn that a Dawson-Watanabe super-process can be recovered from its range. Furthermore, dS can be used to construct a Brownian motion on 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. © Institute of Mathematical Statistics, 2009.