Local times and capacity for transient branching random walks

Abstract

We consider branching random walks on the Euclidean lattice mostly in dimensions five and higher. In this non-Markovian setting, we derive upper bounds for the probability of spending a fixed time in each ball of an arbitrary finite collection of balls. These bounds involve the branching capacity introduced by Zhu (On the critical branching random walk I: branching capacity and visiting probability, arXiv:1611.10324). For random walks, the analogous tail estimates have been instrumental in tackling deviation issues for the volume of the range, and other issues related to excess folding. To obtain these upper bounds, we first obtain a relationship between the equilibrium measure and Green’s function, in the form of an approximate last passage decomposition. Secondly, we obtain exponential moment bounds for functionals of the branching random walk, under optimal condition, analogous to the celebrated Kac’s moment formula for simple random walk. As a corollary we obtain an approximate variational characterisation of the branching capacity.