# > BUC-IX: 11.12.17 - 15.12.17

## Overview

This meeting has been delayed until December 2017 due to last minute issues.

New date: 11th-15th December 2017

The purpose of this meeting is allow for an update in progress between the Mexican and Prob-L@B communities in the field of branching processes and related topics with participation from other international partners. The programme consists of recent results and time for collaborative work as new project fomrulation.

## Organizers

Andreas Kyprianou (University of Bath, UK)
Juan Carlos Pardo (CIMAT, México)

## Courses

 Course Title A Gabriel Berzunza (University of Gottingen) The pruning process on trees Slides Aldous and Pitman introduced a tree-valued Markov chain by pruning off more and more sub-trees along the edges of a Galton-Watson tree.  More generally, Abraham and Delmas defined a non-uniform pruning process on the branch points of a Galton-Watson tree. In the same spirit, some authors have considered the continuum tree analogues (i.e. Lévy trees) of pruning dynamics. The aim of this mini course is to present a topology which allows to link the discrete and continuum tree-valued dynamics. Roughly speaking,  we construct the pruning process on the space of so-called bi-measure trees, which are metric measure spaces with an additional pruning measure. B Sandra Palau (University of Bath) Brownian web and net Slides The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the scaling limits of many one-dimensional interacting particle systems with branching and coalescence. In this course we will study how to construct these processes and some of their properties. C Alex Watson(University of Manchester)  Quan Shi (University of Oxford) Growth-fragmentations and random planar maps Quan Notes Quan Slides Alex Notes   Fragmentation models incorporating growth have a long history in biological applications, but only recently have we been able to study general growth-fragmentation models from a probabilistic viewpoint. The framework of Bertoin and coauthors gives a construction of Markovian growth-fragmentation processes in a very general setting; these processes are interesting in themselves, but they can also be used to describe the size of cycles in certain random planar maps with boundary. In this course, we will discuss the construction and properties of growth-fragmentations, with a focus on the self-similar case, and outline the connection with random planar maps. D Minmin Wang (University of Bath) Scaling limits of inhomogeneous trees and graphs Slides 1 Slides 2 Slides 3 The aim of this mini course is to discuss some recent developments on the scaling limits of critical inhomogeneous graphs. We will see that the critical random graphs we consider are closely related to a family of Galton—Watson trees. Thus, our approach to the problem consists in analysing these Galton—Watson trees and their scaling limits, i.e. Lévy trees introduced by Le Gall & Le Jan. E Adrian Gonzalez (UNAM) The discrete ancestral selection graph Slides In this mini curse we will talk about a random graph approach to population genetics which allow us to understand the relation between forward in time frequency processes (diffusions with jumps) and backwards in time ancestral processes (branching processes with interactions).

## Timetable

 Monday 08:30-09:30 Breakfast 09:30-10:30 A 10:30-10:45 Coffee 10:45-11:45 B 11:45-12:00 Coffee 12:00-12:30 Arno Siri-Jégousse (UNAM): Poisson Dirichlet distributions applied to Coalescents We study three examples where the Poisson Dirichlet distribution for exchangeable partitions provides a straightforward tool to study functionals of coalescent proceses, eith a straightforward application ti population genetics. After recalling the celebrated Ewens' sampling formula for the Kingman coalescent, we will study the minimal clade size and the sites frequency spectrum of the Bolthausen-Sznitman coalescent. 12:30-13:00 Juan Carlos Pardo (CIMAT): Extinction time of a CSBP with competition in a Lévy random environment In this talk, we are interested on the extinction time of continuous state  branching processes with competition  in a Lévy random environment. In particular we prove,  under the so-called Grey's condition together with the assumption that the competition mechanism is  one-sided Lipschitz and the Lévy random environment does not drift towards $+\infty$ ,  that for any starting point the process gets  extinct in finite time a.s. Moreover if we  replace the condition on the Lévy random environment by a technical integrability condition on the competition mechanism, then  the  process also  gets  extinct in finite time a.s. and comes down from infinity. If the random environment is driven by a Brownian motion, we provide a Lamperti-type representation that allow us to get more interesting results regarding the extinction of such models. 13:00-15:00 Lunch 15:00-17:00 Work sessions 17:00-18:00 A 18:00-20:00 Taquiza and fermented horse milk

 Tuesday 08:30-09:30 Breakfast 09:30-10:30 B 10:30-10:45 Coffee 10:45-11:45 A 11:45-12:00 Coffee 12:00-12:30 Cecile Mailler (University of Bath): Multi-drawing multi-colour urns A classical Pólya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_(1\leq i, j \leq d)$. At every discrete time-step, we draw a ball uniformly at random, denote its colour c, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1 \leq j \leq d$). This talk will focus on multi-drawing Pólya urns, where the replacement rule depends on the random drawing of a set of m balls from the urn (with or without replacement). I will show how to apply stochastic approximation methods to get up-to-second-order limit theorems for balanced (but not “affine” urn schemes). This is a joint work with Nabil Lasmar and Olfa Selmi (Monastir, Tunisia). 12:30-13:00 Maria Clara Fitipaldi: Ray-Knight representation of Lévy-driven continuous-state branching processes with competition Abstract: In this talk we talk about flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li, we construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Lévy-trees, and establish a Ray-Knight representation result for these processes in terms of the local times of suitably pruned forests. 13:00-15:00 Lunch 15:00-17:00 Work sessions 17:00-18:00 C

 Wednesday 08:30-09:30 Breakfast 09:30-10:30 B 10:30-10:45 Coffee 10:45-11:45 C 11:45-12:00 Coffee 12:00-13:00 D 13:00-15:00 Lunch 15:00-16:00 C

 Thursday 08:30-09:30 Breakfast 09:30-10:30 C 10:30-10:45 Coffee 10:45-11:45 D 11:45-12:00 Coffee 12:00-12:30 Liz Peñaloza Velasco: The shape of a seed bank tree The propose of this work is study the total lenght of a seed bank coalescent which is used to model the genealogies of population of plants and seeds where seeds can stay inactive for a long period. For statistical applications, we are only interested in the part of plants of the tree because the mutations can only occur on this side.  To do this we define some useful stopping times like the first time that a plant becomes a seed and the first time that a seed becomes a plant, and we study the block counting process at those times. Also, we will use a connection with the Poisson-Dirichlet distribution to calculate the external branches lenght of the seed bank coalescent. 12:30-13:00 Airam Blancas (Frankfurt University):Marked metric spaces for finite populations Consider a finite population of individuals named $i=1,2, .., N$ with genealogical distances $r(i,j)$ and types $g(i)$ . With the distance matrix r and the type vector g, the set $[N]=\{1,..., N\}$ forms a marked metric space (in the sense of Greven, Pfaffelhuber and Winter (2012)). Now think of a population dynamics where the individual reproduction depends on the type composition of the population as well as the individual type. In our talk we will explain why a good state space for the resulting stochastic process consists of equivalence classes of marked metric spaces, where $([N],r, g)$ and  $([N],r', g')$ are considered equivalent if $r(i,j) = r(\pi(i), \pi(j))$ and $g'(i) = g(\pi(i))$ for some permutation $\pi$ of $[N]$ . (In other words, the specific namimg of the individuals should not matter as long as the types and the genelogical distances are respected.) We will put this in action by comparing a lookdown version and a branching particle system version of the population dynamics. These two version result in different processes for the marked metric spaces, but in the same process for their equivalence classes. Based on a joint work with S. Gufler, S. Kliem, V. C. Tran and A. Wakolbinger. 13:00-15:00 Lunch 15:00-17:00 Work sessions 17:00-18:00 E

 Friday 08:30-09:30 Breakfast 09:30-10:30 E 10:30-10:45 Coffee 10:45-11:45 D 11:45-12:00 Coffee 12:00-12:30 Janos Englander (University of Colorado, Boulder):Survival asymptotics for branching random walks in IID environments ABSTRACT: We first study a model, introduced recently in [E and N. Sieben, 2011], of a critical branching random walk in an IID random environment on the d-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no obstacle' placed there. The obstacles appear at each site with probability  $0\le p<1$, independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time n. We show that on a set  of full measure of the obstacle configuration, as n tends to infinity, $P^{\omega}(S_n)$ is asymptotical with $2/(qn)$ in the critical case, while this probability is asymptotically stretched exponential in the subcritical case. Hence, the model exhibits self-averaging' in the critical case but not in the subcritical one. I.e., in the first case, the asymptotic tail behavior is the same as in a `toy model' where space is removed, while in the second, the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. A spine decomposition of the branching process along with  known results on random walks are utilized. This is joint work with Y. Peres (Microsoft Research). 12:30-13:30 E 13:30-15:00 Lunch 15:00-17:00 Work sessions

## Participants

Bath CIMAT UNAM Other
Andreas Kyprianou
Sandra Palau
Emma Horton
Dorottya Fekete
Minmin Wang
Tsogzolmaa Saizmaa
Weerapat Satitkanitkul
Isaac Gonzalez
Cecile Mailler
Juan Carlos Pardo
Victor Rivero
Rodrigo Gachuz
Jose Luis Perez
Antonio Murillo
Jose-Alfredo Mimbela
Ekaterina Todorova
Maria Fernanda Serna Mendoza
Eduardo Alvarez Rodríguez
Jose Alberto Tepox Mendez
Camilo González González
Natalia Cardona Tobón
Santiago Arenas Velilla
Jose de Jesus Contreras Arredondo

Arno Siri-Jégousse
Gerónimo Uribe Bravo