Martin boundaries and asymptotic behavior of branching random walks
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914862111653888 |
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| author | Bertacchi, Daniela Candellero, Elisabetta Zucca, Fabio |
| author_facet | Bertacchi, Daniela Candellero, Elisabetta Zucca, Fabio |
| contents | Let $G$ be an infinite, locally finite graph. We investigate the relation between supercritical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic directions taken by the particles, and as a consequence we find a new connection between $t$-Martin boundaries and standard Martin boundaries. Moreover, given a subgraph $U$ we study two aspects of branching random walks on $U$: when the trajectories visit $U$ infinitely often (survival) and when they stay inside $U$ forever (persistence). We show that there are cases, when $U$ is not connected, where the branching random walk does not survive in $U$, but the random walk on $G$ converges to the boundary of $U$ with positive probability. In contrast, the branching random walk can survive in $U$ even though the random walk eventually exits $U$ almost surely. We provide several examples and counterexamples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_03711 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Martin boundaries and asymptotic behavior of branching random walks Bertacchi, Daniela Candellero, Elisabetta Zucca, Fabio Probability 60J80, 60J10, 60J45 Let $G$ be an infinite, locally finite graph. We investigate the relation between supercritical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic directions taken by the particles, and as a consequence we find a new connection between $t$-Martin boundaries and standard Martin boundaries. Moreover, given a subgraph $U$ we study two aspects of branching random walks on $U$: when the trajectories visit $U$ infinitely often (survival) and when they stay inside $U$ forever (persistence). We show that there are cases, when $U$ is not connected, where the branching random walk does not survive in $U$, but the random walk on $G$ converges to the boundary of $U$ with positive probability. In contrast, the branching random walk can survive in $U$ even though the random walk eventually exits $U$ almost surely. We provide several examples and counterexamples. |
| title | Martin boundaries and asymptotic behavior of branching random walks |
| topic | Probability 60J80, 60J10, 60J45 |
| url | https://arxiv.org/abs/2308.03711 |