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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2501.14255 |
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| _version_ | 1866908582615711744 |
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| author | Lee, Cheuk Yin Xiao, Yimin |
| author_facet | Lee, Cheuk Yin Xiao, Yimin |
| contents | Let $W= \{W(t): t \in \mathbb{R}_+^N \}$ be an $(N, d)$-Brownian sheet and let $E \subset (0, \infty)^N$ and $F \subset \mathbb{R}^d$ be compact sets. We prove a necessary and sufficient condition for $W(E)$ to intersect $F$ with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set $W(E)\cap F$ in terms of the thermal capacity of $E \times F$. This extends the previous results of Khoshnevisan and Xiao (2015) for the Brownian motion and Khoshnevisan and Shi (1999) for the Brownian sheet in the special case when $E \subset (0, \infty)^N$ is an interval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hitting probabilities, thermal capacity, and Hausdorff dimension results for the Brownian sheet Lee, Cheuk Yin Xiao, Yimin Probability Let $W= \{W(t): t \in \mathbb{R}_+^N \}$ be an $(N, d)$-Brownian sheet and let $E \subset (0, \infty)^N$ and $F \subset \mathbb{R}^d$ be compact sets. We prove a necessary and sufficient condition for $W(E)$ to intersect $F$ with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set $W(E)\cap F$ in terms of the thermal capacity of $E \times F$. This extends the previous results of Khoshnevisan and Xiao (2015) for the Brownian motion and Khoshnevisan and Shi (1999) for the Brownian sheet in the special case when $E \subset (0, \infty)^N$ is an interval. |
| title | Hitting probabilities, thermal capacity, and Hausdorff dimension results for the Brownian sheet |
| topic | Probability |
| url | https://arxiv.org/abs/2501.14255 |