Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915700697726976 |
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| author | Chen, Xinxin Lin, Shen |
| author_facet | Chen, Xinxin Lin, Shen |
| contents | We study the critical branching random walk on $\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\|x\|^{4-d}$ in dimensions $d\leq 3$, of order $\log\|x\|$ in dimension $d=4$, and of order 1 in dimensions $d\geq 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$ Chen, Xinxin Lin, Shen Probability 60G55, 60J80 We study the critical branching random walk on $\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\|x\|^{4-d}$ in dimensions $d\leq 3$, of order $\log\|x\|$ in dimension $d=4$, and of order 1 in dimensions $d\geq 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265). |
| title | Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$ |
| topic | Probability 60G55, 60J80 |
| url | https://arxiv.org/abs/2512.24047 |