Boundedness of discounted tree sums

Abstract

Let $(V(u),u\in \mathcal{T})$ be a (supercritical) branching random walk and $(\hat{I}{\cdot }_u,u\in \mathcal{T})$ 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 $\hat{I}¾$ of the tree, we associate the {\it discounted tree sum} $D(\hat{I}¾)$ which is the sum of the $e^{-V(u)}\hat{I}{\cdot }_u$ taken along the ray. The paper deals with the finiteness of ${\sup }_{\hat{I}}¾D(\hat{I}¾)$. To this end, we study the extreme behaviour of the local time processes of the paths $(V(u),u\in \hat{I}¾)$. 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.