Branching random walks and Minkowski sum of random walks
| Autor: | Asselah, Amine, Okada, Izumi, Schapira, Bruno, Sousi, Perla |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension $d\ge 5$, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of $N$ independent simple random walk ranges in $\mathbb{Z}^d$, with $d>2N$, and of a finite set $A\subset \mathbb{Z}^d$. When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of $A$ with respect to the kernel $K(x,y)=(1+\|x-y\|)^{2N-d}$. As a consequence, we establish a new relation between capacity and {\it branching capacity}. Comment: 25 pages |
| Databáze: | arXiv |
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