Support of the limiting boundary measure of branching random walk

Candellero and Hutchcroft proved that for a supercritical branching random walk on a transient irreducible Markov chain, with an offspring law satisfying the Llog L condition, the normalised occupation measures converge almost surely to a random measure W on the Martin boundary, and asked for conditions under which the support of W coincides almost surely with the set of accumulation points of the walk. We show that this holds whenever the boundary has a countable basis of open sets V such that the exit probability Pₓ(X_∞∈ V) stays bounded away from zero as x tends to any point of V. The condition holds whenever the Martin compactification is Dirichlet regular. It therefore covers simple random walk on regular trees, for which we give a self-contained proof, and finitely supported symmetric random walks on non-elementary hyperbolic groups, for every supercritical Llog L offspring law. The proof needs only the integrability of the martingale limit, with no second moment, and it applies on both sides of the local survival threshold.