2019/09/05 by Noah Forman, Forman, Noah, Soumik Pal +5
Biochemistry, Genetics and Molecular Biology · Mathematics · #60G18 #60G52 #60G55 #60J25 #60J60 #60J80 #Diffusion and Search Dynamics #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1909.02584
openalex publication_date 2019/09/05 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
Consider a spectrally positive Stable(1+\α) process whose jumps we\ninterpret as lifetimes of individuals. We mark the jumps by continuous\nexcursions assigning "sizes" varying during the lifetime. As for\nCrump-Mode-Jagers processes (with "characteristics"), we consider for each\nlevel the collection of individuals alive. We arrange their "sizes" at the\ncrossing height from left to right to form an interval partition. We study the\ncontinuity and Markov properties of the interval-partition-valued process\nindexed by level. From the perspective of the Stable(1+\α) process, this\nyields new theorems of Ray-Knight-type. From the perspective of branching\nprocesses, this yields new, self-similar models with dense sets of birth and\ndeath times of (mostly short-lived) individuals. This paper feeds into projects\nresolving conjectures by Feng and Sun (2010) on the existence of certain\nmeasure-valued diffusions with Poisson--Dirichlet stationary laws, and by\nAldous (1999) on the existence of a continuum-tree-valued diffusion.\n