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Hausdorff measure of arcs and Brownian motion on Brownian spatial trees. (English) Zbl 1219.60052

Summary: A Brownian spatial tree is defined to be a pair \(({\mathcal T},\varphi)\), where \(\mathcal T\) is the rooted real tree naturally associated with a Brownian excursion and \(\varphi \) is a random continuous function from \(\mathcal T\) into \(\mathbb R^d\) such that, conditional on \({\mathcal T}, \varphi \) 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 \({\mathcal S}: = \varphi ({\mathcal T})\). Applications of this result include the recovery of the spatial tree \(({\mathcal T}, \varphi)\) 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.

MSC:

60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
28A78 Hausdorff and packing measures
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