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Branching capacity and Brownian snake capacity
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abstract
The branching capacity has been introduced by [Zhu 2016] as the limit of the hitting probability of a symmetric branching random walk in $\mathbb Z^d$, $d\ge 5$. Similarly, we define the Brownian snake capacity in $\mathbb R^d$, as the scaling limit of the hitting probability by the Brownian snake starting from afar. Then, we prove our main result on the vague convergence of the rescaled branching capacity towards this Brownian snake capacity. Our proof relies on a precise convergence rate for the approximation of the branching capacity by hitting probabilities.
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Cited by 1 Pith paper
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Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$
Conditioned on hitting a distant set K, the total occupation time of a critical branching random walk is of order ||x||^{4-d} for d≤3, log||x|| for d=4, and bounded for d≥5, with explicit weak limits in all dimensions.
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