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We consider branching random walks on the Euclidean lattice in dimensionsfive and higher. In this non-Markovian setting, we first obtain a relationshipbetween the equilibrium measure and Green's function, in the form of anapproximate last passage decomposition. Secondly, we obtain exponential momentbounds for functionals of the branching random walk, under optimal condition.As a corollary we obtain an approximate variational characterisation of thebranching capacity. We finally derive upper bounds involving the branchingcapacity for the tail of the time spent in an arbitrary finite collection ofballs. This generalises the results of [AHJ] and [AS22] for $d\geq 5$. Forrandom walks, the analogous tail estimates have been instrumental tools fortackling deviations problems on the range, related to folding of the walk.

Local times and capacity for transient branching random walks