Boundedness of discounted tree sums

Let (V(u),\, u∈ T) be a (supercritical) branching random walk and (ηᵤ,\,u∈ T) be marks on the vertices of the tree, distributed in an i.i.d.\ fashion. Following Aldous and Bandyopadhyay {AB05}, for each infinite ray ξ of the tree, we associate the { discounted tree sum} D(ξ) which is the sum of the e-V(u)ηᵤ taken along the ray. The paper deals with the finiteness of _ξ D(ξ). To this end, we study the extreme behaviour of the local time processes of the paths (V(u),\,u∈ ξ). It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay {AB05}. We also present several open questions.