dorsal/arxiv
View SchemaLocal times and excursions for self-similar Markov trees
| Authors | Jean Bertoin, Armand Riera, Alejandro Rosales-Ortiz |
|---|---|
| Categories | |
| ArXiv ID | 2601.10610vv1 |
| URL | https://arxiv.org/abs/2601.10610 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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.
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"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 $x\u003e0$ 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\\\u0027evy tree.",
"arxiv_id": "2601.10610",
"authors": [
"Jean Bertoin",
"Armand Riera",
"Alejandro Rosales-Ortiz"
],
"categories": [
"math.PR"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Local times and excursions for self-similar Markov trees",
"url": "https://arxiv.org/abs/2601.10610",
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