INTERVAL FRAGMENTATIONS WITH CHOICE: EQUIDISTRIBUTION AND THE EVOLUTION OF TAGGED FRAGMENTS

• 成果类型: Article
• 署名作者: Maillard, Pascal; Paquette, Elliot
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; McGill University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1766
• 发表日期: 2022
• 页码: 3537-3571
• 关键词: stability

摘要

We consider a Markovian evolution on point processes, the psi -process, on the unit interval in which points are added according to a rule that depends only on the spacings of the existing point configuration. Having chosen a spacing, a new point is added uniformly within it. Building on previous work of the authors and of Junge, we show that the empirical distribution of points in such a process is always equidistributed under mild assumptions on the rule, generalizing work of Junge. A major portion of this article is devoted to the study of a particular growth-fragmentation process, or cell process, which is a type of piecewise-deterministic Markov process (PDMP). This process represents a linearized version of a size-biased sampling from the psi-process. We show that this PDMP is ergodic and develop the semigroup theory of it, to show that it describes a linearized version of the psi-process. This PDMP has appeared in other contexts, and in some sense we develop its theory under minimal assumptions.