摘要
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 （CMJ） 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.
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 （CMJ） 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.
基金
Hui He wanted to thank Concordia University for his pleasant stay at Montreal where this work was done. The authors would like to thank Professor Wenming Hong for his enlightening discussion. They also would like to thank Amaury Lambert for suggesting the time reversal treatment of the model in Section 5. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11201030, 11371061）, the Fundamental Research Funds for the Central Universities （2013YB59）, and the Natural Sciences and Engineering Research Council of Canada.
