  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.
