Local times and excursions for self-similar Markov trees

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.