Random fragmentation-coalescence processes out of equilibrium

随机聚并模型描述了质量块如何根据随机进化的某些规则随时间随机结合在一起。相反，随机破碎模型描述了质量块如何随着时间的推移而分裂，同样是根据随机进化的某些规则。具有上述一种或另一种作用的过程已被广泛研究。自斯摩鲁霍夫斯基100年前的开创性工作以来，合并一直是一个活跃的领域。碎片化是最近才被研究的现象，主要的基础发展始于21世纪初Bertoin的工作。然而，在严谨的数学文献中，关于分裂和合并这两种作用的结合，几乎没有做过什么。尽管如此，由于在物理化学和家谱中的应用，以及最近在社会科学和生物学中的群体动力学中的应用，在科学文献中有很强的动机来处理这些模型。该项目的目的是研究新的概率技术，以表征随机破碎-合并过程的可处理家族的动力学。这种模型的数学困难之一是它们不具有所谓的可逆性。这意味着，当考虑这些过程的时间反转时，它们不会表现出允许使用已知分析技术的数学便利性。由于这个原因，他们的分析通常是困难的。在本提案中，我们将研究一些特殊类别的碎片-合并模型，这些模型最近才被引入文献（由PI和CI以及其他人），并且已经证明了某种程度的可追溯性。我们将使用混合技术来分析它们的平稳和准平稳行为，暴露目前未知的行为，并对如何处理这些模型有更深入的了解。

Stochastic coalescence models describe how blocks of mass randomly join together over time according certain rules of random evolution. Conversely, stochastic fragmentation models describe how blocks of mass break apart over time, again according to certain rules of random evolution. Processes with one or antoher of theses actions, have been widely investigated. Coalescence has been an active field since the seminal work of Smoluchowski 100 years ago. Fragmentation is a more recently investigated phenomenon, with the the main foundational development starting from the work of Bertoin in the early 2000s. Little has been done, howver, in the rigorous mathematical literature regarding the combination of both actions of fragmentation and coalescence. Despite this fact, there is a strong motivation for the treatement of such models in the scientific literature thanks to applications in physical chemistry and genealogy, and more recently in group dynamics in the social sciences and biology.The purpose of this project is to thus investigate new probabilistic techniques to characterise the dynamics of tractable families of stochastic fragmentation-coalescence processes. One of the mathematical difficulties with such models is that they do not possess so-called reversibility properties. This means that when considering such processes time reversed, they do not exibit the mathematical convenience that would allow known analytical techniques to be used. For this reason, their analysis is generally difficult.In this proposal we will look at some special classes of fragmentation-coalescence models that were only very recently introduced into the literature (by the PI and CI as well as others) and for which some degree of tractability has already been demonstrated. We will use a mixture of techniques to analyse their stationary and quasi-stationary behaviour, exposing currently unknown behaviours and laying out a deeper understanding of how such models can be treated in general.