IS19: Branching and Interacting Particle Systems

Organizer: Dariusz Buraczewski & Piotr Dyszewski (University of Wrocław)

Discounted tree sums in branching random walks

YUEYUN HU

This talk is based on a joint work with Elie Aïdékon and Zhan Shi. 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 (2005), for each infinite ray \(\xi\) of the tree, we associate the 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 (2005). We also present several open questions.

Explosion of Crump-Mode-Jagers processes with critical immediate offspring

Matthias Meiners

In my talk, I will consider the phenomenon of explosion in general (Crump-Mode-Jagers) branching processes, which refers to the event where an infinite number of individuals are born in finite time. In a critical setting where the expected number of immediate offspring per individual is exactly 1, whether or not explosion occurs depends on the fine properties of the reproduction point process. I will review some known results and explain recent results in this area. In particular, I present a necessary and sufficient condition in the case where the reproduction point process is Poissonian.