Diffusions on a space of interval partitions: construction from marked L\'evy processes

TL;DR

This paper constructs a new class of measure-valued diffusions using stable Lévy processes and interval partitions, revealing novel Markov and continuity properties and contributing to longstanding conjectures in stochastic processes.

Contribution

It introduces a new framework linking stable Lévy processes with interval partitions, providing novel theorems and models for measure-valued diffusions and continuum trees.

Findings

Established Markov and continuity properties of the interval-partition process

Derived new Ray-Knight-type theorems for stable processes

Developed self-similar models with dense birth-death times

Abstract

Consider a spectrally positive Stable($1+\alpha$) process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning "sizes" varying during the lifetime. As for Crump-Mode-Jagers processes (with "characteristics"), we consider for each level the collection of individuals alive. We arrange their "sizes" at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable($1+\alpha$) process, this yields new theorems of Ray-Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson--Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.