Tree-valued random processes: tree growth, pruning and stationary processes in spaces of continuum trees

随机的树木和森林自然地出现在许多生物和物理模型中。这包括系统发育和更一般的谱系树，聚集和破碎模型中的碰撞/分割历史，以及簇/界面/晶体生长模型中的形状。这导致了离散和连续框架中的各种数学模型。最近的研究已经转移到包含额外结构的树木，如空间运动，测量，点过程或区分顶点，经历生长，减少或重组的动态进化的树木，以及将树木作为研究更丰富的组合和连续结构（如平面图和稀疏图）的工具。本项目提出的研究目的是为这些模型的文献做出贡献，特别强调树值随机过程。虽然大多数目标是回答概率学家感兴趣的自然问题，但这些方法将需要与连续体树空间的几何和拓扑相关的一些发展。David Aldous在1991- 1993年发表了他关于（布朗）连续统随机树（CRT）的开创性著作。在接下来的20年里，人们对这些最初的概念进行了大量的研究、概括、关联和应用。布朗运动是具有足够独立性和有限方差（控制大跳跃）的离散随机过程的通用尺度极限，布朗CRT是具有足够独立性和有限方差（控制顶点度）的离散树的通用尺度极限。为了使离散树收敛到一个尺度极限的数学意义，Aldous将度量树嵌入到可和序列的空间中。Evans和他的合作者考虑了紧凑的树状度量空间的等距类集合，几何学家和其他人已经发展了一个由Gromov-Hausdorff拓扑诱导的收敛概念。这些思想已被扩展到处理有根树、加权树和局部紧凑树。当树也具有平面顺序，并且权重度量的支持在树上很密集时，树结构可以用高度函数进行规范编码，从而获得更强的收敛形式。对于具有n个顶点/叶子的均匀树，有三种动态导致了连续统树的动力学：重新嫁接、子树修剪和重新嫁接的根生长Aldous- broder算法，以及现在被称为连续统模拟中的Aldous扩散的叶子移除和重新附着算法。它们是三个树值马尔可夫过程，它们的平稳分布是布朗CRT的分布。对非均匀特征在大n下持续存在的模型的研究自然导致了更一般的crt类别。这类crt中最突出的两类是Levy树和自相似/碎片crt，它们的交集由一个单参数的稳定crt类组成。这些crt通常具有，几乎可以肯定，密集的叶集和密集的无限次分支点集。对于所有这些crt，有各种各样的马尔可夫生长和缩小程序。该研究计划的主要目的是构建和研究连续统树空间中的值的马尔可夫过程，这些连续统树的分布具有自相似的crt或Levy树的分布作为其平稳分布。对列维树和自相似crt的研究现在已经达到了一个阶段，许多性质和操作的出现使得这项研究非常及时。尽管二元布朗CRT的例子是简单离散算法的类似物，但已经有相当多的技术努力，自相似CRT和Levy树的更精细的分支结构使我们提出的系统研究在更一般的框架下是一个具有挑战性但可行的项目，具有足够的空间进行令人兴奋的进一步发展。

Random trees and forests appear naturally in many biological and physical models. This includes phylogenetic and more general genealogical trees, collision/partition histories in aggregation and fragmentation models as well as shapes in cluster/interface/crystal growth models. This has led to a variety of mathematical models in both discrete and continuum frameworks. Recent research has moved to trees incorporating additional structures such as spatial motions, measures, point processes or distinguished vertices, to trees undergoing a dynamical evolution of growth, reduction or restructuring, and to trees as tools to study richer combinatorial and continuum structures such as planar maps and sparse graphs. The aim of the research proposed for this project is to contribute to the literature on such models, with a particular emphasis on tree-valued random processes. While most goals are to answer natural questions of interest to probabilists, the methods will require some developments related to the geometry and topology of spaces of continuum trees.David Aldous published his seminal work on the (Brownian) Continuum Random Tree (CRT) in 1991-93. Over the following 20 years, there has been a huge amount of work studying, generalising, relating and applying these initial concepts. Like Brownian motion is the universal scaling limit of discrete random processes with enough independence and finite variance (controlling large jumps), the Brownian CRT appears as the universal scaling limit of discrete trees with enough independence and finite variance (controlling vertex degrees).To make mathematical sense of a convergence of discrete trees to a scaling limit, Aldous used embeddings of metric trees into the space of summable sequences. Evans and co-authors considered the set of isometry classes of compact tree-like metric spaces, for which geometers and others have developed a notion of convergence induced by a Gromov-Hausdorff topology. These ideas have been extended to deal with rooted trees, weighted trees and locally compact trees. When trees are also equipped with a planar order and the support of the weight measure is dense on the tree, the tree structure can be canonically encoded by a height function, giving access to stronger forms of convergence.Three dynamics for uniform trees with n vertices/leaves have given rise to dynamics on continuum trees: the Aldous-Broder algorithm of root-growth with re-grafting, subtree-prune and re-graft and a leaf-removal and re-attachment algorithm now known as Aldous's diffusion in the continuum analogue. They are three tree-valued Markov processes that have as their stationary distribution the distribution of the Brownian CRT.The study of models where non-uniform features persist for large n have naturally led to more general classes of CRTs. The two most prominent classes of such CRTs are Levy trees and self-similar/fragmentation CRTs, whose intersection consists of a one-parameter class of stable CRTs. These CRTs typically have, almost surely, a dense set of leaves and a dense set of branch points of infinite degree. For all these CRTs, there is a variety of Markovian growth and reduction procedures.The principal aim of the research programme for this grant is to construct and study Markov processes with values in a space of continuum trees that have the distributions of self-similar CRTs or Levy trees as their stationary distribution. The study of Levy trees and self-similar CRTs is now reaching a stage where many properties and operations emerge that make this research very timely. Even though the examples for the binary Brownian CRT are analogues of simple discrete algorithms, there has been considerable technical effort, and the more delicate branching structure of self-similar CRTs and Levy trees make our proposed systematic study in the more general framework a challenging but feasible project with ample scope for exciting further developments.