Random fragmentation-coalescence processes out of equilibrium

随机聚结模型描述了质量块如何根据随机演化的某些规则随时间随机结合在一起。相反，随机碎裂模型描述了质量块如何随着时间的推移而分裂，同样是根据随机演化的某些规则。具有这些动作中的一个或更多的过程已被广泛研究。自100年前斯莫卢乔夫斯基的开创性工作以来，合并一直是一个活跃的领域。碎片化是一个较新的研究现象，其主要的基础发展始于本世纪初贝尔托因的工作。然而，在严格的数学文献中，关于碎裂和结合这两种作用的组合，几乎没有做过什么。尽管如此，由于在物理化学和系谱中的应用，以及最近在社会科学和生物学中的群体动力学方面的应用，科学文献中有一个强烈的动机来处理这样的模型。因此，这个项目的目的是研究新的概率技术来表征可处理的随机碎裂-合并过程家族的动力学。这类模型的数学困难之一是它们不具备所谓的可逆性。这意味着，当考虑这种时间倒转的过程时，它们不会表现出允许使用已知分析技术的数学方便性。由于这个原因，他们的分析通常是困难的。在这项建议中，我们将着眼于一些特殊的破碎-合并模型，这些模型最近才被引入文献(由PI和CI以及其他人引入)，并且已经证明了某种程度的易操纵性。我们将使用混合技术来分析它们的静态和准静态行为，揭示目前未知的行为，并对如何从总体上处理此类模型有更深的理解。

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.