Branching Particle Systems in Spectrally One-sided Levy Processes

TL;DR

This paper models a branching particle system derived from spectrally one-sided Levy processes, linking excursion theory to branching structures and measure-valued processes, with applications to understanding complex stochastic dynamics.

Contribution

It introduces a novel construction of a branching particle system from Levy process excursions, connecting nonlocal branchings with measure-valued Markov processes.

Findings

Constructed a branching particle system from Levy excursions.

Linked the total mass to Crump-Mode-Jagers branching process.

Extended results to spectrally negative Levy processes via time reversal.

Abstract

We investigate the branching structure coded by the excursion above zero of a spectrally positive Levy process. The main idea is to identify the level of the Levy excursion as the time and count the number of jumps upcrossing the level. By regarding the size of a jump as the birth site of a particle, we construct a branching particle system in which the particles undergo nonlocal branchings and deterministic spatial motions to the left on the positive half line. A particle is removed from the system as soon as it reaches the origin. Then a measure-valued Borel right Markov process can be defined as the counting measures of the particle system. Its total mass evolves according to a Crump-Mode-Jagers branching process and its support represents the residual life times of those existing particles. A similar result for spectrally negative Levy process is established by a time reversal approach. Properties of the measure-valued processes can be studied via the excursions for the corresponding Levy processes.