# > BUC-XIV: 20.08.18 - 24.08.18

#### Probability challenges

The BUC will centre around a collection of short courses earlier in the week. During the short courses, some tractable problems will be offered to PhD students who will be given on the Tuesday. Students have until Friday to solve in groups and present, thereby becoming better familiar with the material.

## Organizers

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

## Courses

 Course Title and Abstract A Mariana Olvera-Cravioto (Berkeley) Slides day 1 Slides day 2 Slides day 3 Slides day 4 Slides day 5 Branching Distributional Equations and their Applications This course will be centered around the theory and applications of distributional equations of the form: $R \stackrel{D}{=} \Phi( N, Q, {C_i}, {R_i})$ where the $\{R_i\}$ are i.i.d. copies of $R$, independent of the vector $(N, Q, {C_i})$, and $\Phi$ is a deterministic map. Such equations arise in a wide range of settings, ranging from the analysis of computer algorithms (e.g. Quicksort and PageRank) and the study of queueing networks with synchronization, to applications in biology and statistical physics. Although the solutions to these equations are not in general unique, we are often interested in one particular solution, the so-called attracting endogenous solution. These special solutions can be constructed on a mathematical structure known as a weighted branching process, and their asymptotic behavior can be explicitly computed in many important cases. To complement the theory, we will also introduce a Monte Carlo algorithm based on the bootstrap that can be effectively used for most choices of $\Phi$ In terms of applications, the course will focus on two specific cases, the linear equation (known as the smoothing transform) and the maximum equation (known as the high-order Lindley equation). The former appears in the analysis of Quicksort and Google’s PageRank, while the latter in the analysis of queueing networks with synchronization. The PageRank example in particular will allow us to explore in more detail the connection between branching distributional equations and complex networks. In particular, we will discuss two different types of random graph models that, due to their local tree-like structure, allow us to make the connection between the graph and a (weighted) branching process precise. The coupling techniques involved lie at the core of much or random graph theory, and are therefore of independent interest. B Andreas Kyprianou (University of Bath) Slides Self-similar Markov processes We will will reconsider the class of stable processes treated as a self-similar Markov process. This means examining their paths through thte lens through two fundamental transformations: (1) The Lamperti Transformation (2) The Riesz-Bogdan-Zak transfomration. We show how the classical theory of Levy processes feeds in an efficeint way into the theory of self-simiar Markov processes more generally. C Elie Aidekon (Paris VI)  Photos of white boards day 1 Photos of white boards day 2 Photos of white boards day 3 Photos of white boards day 4 Photos of white boards day 5 From branching random walks to nested conformal loops We will see how we can use branching random walk technics to study extreme nesting in the setting of collections of nested loops in the plane which are conformally invariant. The motivation of such a work comes from the construction of the planar Gaussian free field via CLE_4 by Miller and Sheffield. It gives an alternative construction of the exponential of the Gaussian free field (see Aru, Powell, Sepulveda). It will be based on the paper « The extremal process for nested conformal loops » available at http://www.lpma-paris.fr/dw/doku.php?id=users:[aidekon]:index. D Cécile Mailler(Bath)  Photo of white boad Lecture 1 Photo of white board Lecture 2 Pólya urns, mutli-type branching processes and smoothing equations. In this mini-course, I will survey a few results about P&oaccent;loya urns; starting with the almost sure convergence of the original urn of Pólya and Eggenberger, the course will then focus on ``irreducible'' Pólya urns. We will use martingales to show almost sure convergence of the composition of the urn when time tends to infinity, show that Pólya urns can be embedded into continuous time and that their embeddings are multi-type branching processes, and show how smoothing equations can be used to study their asymptotic behaviour, both in discrete and continuous time.

## Timetable

 Monday 08:30-09:30 Breakfast 09:30-10:30 A 10:30-11:30 B 11:30-12:00 Coffee 12:00-13:00 C 13:00-14:30 Lunch 14:30-15:30 Arno Siri-Jegousse Simple nested coalescents Notes 15:30-18:00 Work sessions with coffee 18:00-20:00 Welcome on the balcony

 Tuesday 08:30-09:30 Breakfast 09:30-10:30 B 10:30-11:30 C 11:30-12:00 Coffee 12:00-13:00 D 13:00-14:30 Lunch 14:30-15:30 A 15:30-16:00 Kei Noba "Generalized scale functions and properties of refracted processes" Slides 16:00-18:00 Work sessions with coffee

 Wednesday 08:30-09:30 Breakfast 09:30-10:30 C 10:30-11:30 A 11:30-12:00 Coffee 12:00-13:00 D 13:00-14:30 Lunch 14:30-15:30 B 15:30-18:00 Work sessions with coffee

 Thursday 08:30-09:30 Breakfast 09:30-10:30 C 10:30-11:30 A 11:30-12:00 Coffee 12:00-13:00 D 13:00-14:30 Lunch 14:30-15:30 B 15:30-16:30 Victor Rivero "Stable processes in Cones" 16:30-18:00 Work sessions with coffee

 Friday 08:30-09:30 Breakfast 09:30-10:30 A 10:30-11:30 B 11:30-12:00 Coffee 12:00-13:00 C 13:00-14:30 Lunch 14:30-15:30 Simon Harris "The genealogy of uniform sample from a Galton Watson process" Slides 15:30-18:00 Solution presentations with coffee Whiteboard photos

## Participants

Bath CIMAT UNAM Other
Andreas Kyprianou
Sandra Palau
Emma Horton
Dorottya Fekete
Minmin Wang
Isaac Gonzalez
Cecile Mailler
Alice Callgero
Juan Carlos Pardo
Victor Rivero
Hélene Leman
Natalia Cardona Tobón
Camilo González
Jose Luis Perez
Antonio Murillo

Arno Siri-Jegousse
Lizbeth Peñaloza Velasco
Alejandro Hernández Wences
Pablo Jorge Hernández Hernández
Miriam Ramírez García
Osvaldo Angtuncio Hernández

Simon Harris
Elie Aidekon
Mariana Olvera-Cravioto
Kei Noba