Reinforced stochastic processes: theory and applications

This project is a discrete probability project on reinforced processes. These processes have a wide range of
applications to, e.g., networks, genetics, and biology; a famous example is the preferential attachment random graph,
which is used as a good model for large, complex, real-life networks such as the internet or social networks.
The project will focus on Pólya urns, which are the building blocks of all reinforced processes. A Pólya urn is a
Markov chain describing the contents of an urn that contains balls of different colours. The original Pólya urns
evolves as follows: at every (discrete) time-step, we draw a ball from the urn uniformly at random, and replace it in
the urn together with an additional ball of the same colour. Many generalisations have been studied, in particular
different "replacement rules", and the goal is to prove asymptotic theorems (law of large numbers, central limit
theorem) for the content of the urn at large time.
Although Pólya urns are classical objects in probability theory and have been studied for a hundred years, new
recent developments have raised numerous open problems with exciting potential applications. For example, Pólya
urns with infinitely-many colours were introduced in 2017 by Mailler and collaborators; as well as being interesting
from a purely theoretical point of view, proving asymptotic theorems about these urns gives a route to obtain exciting
information about other stochastic objects such as random trees.
The PhD will focus on proving limit theorems about branching Markov chains on the random recursive tree (the
generalisation of Pólya urns to infinitely-many colours); the first question to be tackled is about the position of the
rightmost particle in this model.