The Hausdorff measure of stable trees

TL;DR

This paper investigates the Hausdorff measure properties of stable trees, including the continuum random tree and its level sets, revealing their measure-theoretic structure and relationships with local time measures.

Contribution

It derives the exact Hausdorff measure function for Aldous' continuum random tree and its level sets, advancing understanding of their geometric measure properties.

Findings

Exact Hausdorff measure function for Aldous' continuum random tree

Equivalence of uniform and local time measures with Hausdorff measures

Results extend to general stable trees with less precision

Abstract

We study fine properties of the so-called stable trees, which are the scaling limits of critical Galton-Watson trees conditioned to be large. In particular we derive the exact Hausdorff measure function for Aldous' continuum random tree and for its level sets. It follows that both the uniform measure on the tree and the local time measure on a level set coincide with certain Hausdorff measures. Slightly less precise results are obtained for the Hausdorff measure of general stable trees.