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Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

2009
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Annals of Probability
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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

doi:10.1214/08-aop425
fatcat:pk7uhihdbrhu3eu7al6mk52udm