Conférence annuelle du GDR branchement

Discounted tree sums in branching random walks.

29 janv. 2025, 11:30

45m

Orsay

Orateur

M. Yueyun Hu

Description

This talk is based on a joint work with Eile Aïdékon and Zhan Shi. Let $(V(u),u\in T)$ be a (supercritical) branching random walk and $({\eta }_u,u\in T)$ be positive marks on the vertices of the tree, distributed in an i.i.d. fashion. Following Aldous and Bandyopadhyay (2005), 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. We take interest in the finiteness of $\underset{\xi }{sup}D(\xi )$. 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.