Boundedness of discounted tree sums

Let (V (u), \, u T) be a (supercritical) branching random walk and (ᵤ, \, 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 of the tree, we associate the discounted tree sum D () which is the sum of the e^-V (u) ᵤ taken along the ray. The paper deals with the finiteness of _ D (). To this end, we study the extreme behaviour of the local time processes of the paths (V (u), \, u). It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay AB05. We also present several open questions.