Branching particle systems in spectrally one-sided Lévy processes

We investigate the branching structure coded by the excursion above zero of a spectrally positive Lévy process. The main idea is to identify the level of the Lévy 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 (CMJ) branching process and its support represents the residual life times of those existing particles. A similar result for spectrally negative Lévy process is established by a time reversal approach. Properties of the measure-valued processes can be studied via the excursions for the corresponding Lévy processes.