Packing and Hausdorff measures of stable trees

TL;DR

This paper investigates the Hausdorff and packing measures of stable trees, a class of random continuous trees, revealing the absence of exact measures for level sets and non-Brownian cases with specific gauge functions.

Contribution

It extends previous work by analyzing Hausdorff and packing measures of stable trees and their level sets, showing the non-existence of exact measures in certain cases.

Findings

No exact packing measure for level sets.

Non-Brownian stable trees lack exact Hausdorff measure with regularly varying gauge functions.

Results generalize previous findings on Brownian and stable trees.

Abstract

In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random tree (1991) which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from a joint work with J-F Le Gall (2006).