Boundedness of discounted tree sums

TL;DR

This paper investigates the conditions under which the supremum of discounted sums along infinite rays in a branching random walk remains finite, analyzing the extremal behavior of associated local time processes.

Contribution

It provides a partial solution to an open problem about the boundedness of discounted tree sums and explores the extremal behavior of local time processes in branching random walks.

Findings

Identifies conditions for finiteness of the supremum of discounted sums.

Analyzes the extremal behavior of local time processes.

Answers a question posed by Nicolas Curien.

Abstract

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.