Abstract

The branching capacity has been proven by Zhu (2016) as the limit of the hitting probability of a symmetric branching random walk in Z^{d, d ≥ 5. Similarly, we define the Brownian snake capacity in R}d, as the scaling limit of the hitting probability on the Brownian snake starting from afar. We prove the vague convergence of the rescaled branching capacity towards this Brownian snake capacity. As an interesting example, we study the branching capacity of the range of a random walk in Z^d. Schapira (2024) has recently obtained precise asymptotics in the case d ≥ 6 and has demonstrated a transition at dimension d = 6. We are interested in the case d = 5 and prove that the renormalized branching capacity converges to the Brownian snake capacity of the range of a Brownian motion.   Co-authors: names and affiliations: Tianyi Bai (AMSS) and Jean-François Delmas (CERMICS)