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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.10610 |
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| _version_ | 1866911377716674560 |
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| author | Bertoin, Jean Riera, Armand Rosales-Ortiz, Alejandro |
| author_facet | Bertoin, Jean Riera, Armand Rosales-Ortiz, Alejandro |
| contents | 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 $μ$ on the tree. Finally, we point out that using a local time measure instead of the usual length measure $λ$ 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 Lévy tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10610 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local times and excursions for self-similar Markov trees Bertoin, Jean Riera, Armand Rosales-Ortiz, Alejandro Probability 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 $μ$ on the tree. Finally, we point out that using a local time measure instead of the usual length measure $λ$ 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 Lévy tree. |
| title | Local times and excursions for self-similar Markov trees |
| topic | Probability |
| url | https://arxiv.org/abs/2601.10610 |