Boundedness of discounted tree sums

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