Local times and excursions for self-similar Markov trees

TL;DR

This paper develops local time measures for self-similar Markov trees, enabling detailed analysis of the decoration process and its excursions, with convergence results linking local times to harmonic measure and implications for tree structure.

Contribution

It constructs local time measures on self-similar Markov trees, characterizes the decoration process along branches, and shows how local times relate to harmonic measure and tree geometry.

Findings

Local time measures are constructed for various levels of the tree.

The law of the decoration along a branch is characterized by a self-similar Markov process.

Normalized local times converge to harmonic measure as the level approaches zero.

Abstract

This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to $[0,\infty)$ that we call the decoration. Here, we construct local time measures $L(x,dt)$ at every level $x>0$ of the decoration for a large class of self-similar Markov trees. This enables us to mark at random a typical point in the tree at which the decoration is $x$. We identify the law of the decoration along the branch from the root to this tagged point in terms of a remarkable (positive) self-similar Markov process. We also show that after a proper normalization, $L(x,dt)$ converges as $x\to 0+$ to the harmonic measure $\mu$ on the tree. Finally, we point out that using a local time measure instead of the usual length measure $\lambda$ to compute distances on the tree turn the latter into a continuous branching tree. This is relevant to analyze the excusions of the decoration away from a given level. Many results of the present work shall be compared with the recent ones in [22,23] about local times and excursions of a Markov process indexed by L\'evy tree.