On the critical branching random walk in supercritical and critical dimensions

We extend several results in the potential theory of random walk to critical branching random walk. In the supercritical dimensions ($$d\geq 5$$), we introduce branching capacity for any finite subset of $$\mathbb{Z}^d$$ and establish its connections with the hitting probability by branching random walk and branching recurrence. In the critical dimension ($$d=4$$), we also establish the asymptotics of the hitting probability and some related results.