INFINITELY RAMIFIED POINT MEASURES AND BRANCHING LEVY PROCESSES

• 成果类型: Article
• 署名作者: Bertoin, Jean; Mallein, Bastien
• 署名单位: University of Zurich; Universite Paris 13
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1292
• 发表日期: 2019
• 页码: 1619-1652
• 关键词: martingales

摘要

We call a random point measure infinitely ramified if for every n is an element of N, it has the same distribution as the nth generation of some branching random walk. On the other hand, branching Levy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some Levy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and Levy processes: the value at time 1 of a branching Levy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time 1 of some branching Levy process.