Reinforced stochastic processes: theory and applications

该方案是关于强化过程的离散概率方案。这些过程在网络、遗传学和生物学中有着广泛的应用；一个著名的例子是优先连接随机图，它被用作大型、复杂的真实网络(如互联网或社交网络)的良好模型。该项目将专注于Pólya骨灰盒，它是所有增强过程的构建块。Pólya骨灰盒是描述骨灰盒内容物的马尔可夫链，该骨灰盒包含不同颜色的球。最初的Pólya骨灰盒如下：在每个(离散的)时间步，我们从骨灰盒中均匀随机地抽出一个球，并用另一个相同颜色的球将其替换在骨灰盒中。虽然Pólya骨灰盒是概率论中的经典对象，并且已经被研究了一百年，但最近的发展提出了许多具有令人兴奋的潜在应用的公开问题。例如，具有无限多个颜色的Pólyaurn是由Mailler和他的合作者在2017年引入的；从纯理论的角度来看，有趣的是证明关于这些骨灰盒的渐近定理给出了一条获得关于其他随机对象(如随机树)的兴奋信息的途径。Phd将专注于证明关于随机递归树上分支马尔可夫链的极限定理(将Pólya骨灰盒推广到无限多个颜色)；第一个要解决的问题是关于这个模型中最大粒子的位置。

This project is a discrete probability project on reinforced processes. These processes have a wide range ofapplications 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 aMarkov chain describing the contents of an urn that contains balls of different colours. The original Pólya urnsevolves as follows: at every (discrete) time-step, we draw a ball from the urn uniformly at random, and replace it inthe urn together with an additional ball of the same colour. Many generalisations have been studied, in particulardifferent "replacement rules", and the goal is to prove asymptotic theorems (law of large numbers, central limittheorem) 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, newrecent developments have raised numerous open problems with exciting potential applications. For example, Pólyaurns with infinitely-many colours were introduced in 2017 by Mailler and collaborators; as well as being interestingfrom a purely theoretical point of view, proving asymptotic theorems about these urns gives a route to obtain excitinginformation 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 (thegeneralisation of Pólya urns to infinitely-many colours); the first question to be tackled is about the position of therightmost particle in this model.