Branching processes in Levy processes: The exploration process

• 成果类型: Article
• 署名作者: Le Gall, JF; Le Jan, Y
• 署名单位: Sorbonne Universite; Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 213-252
• 关键词: brownian excursion; random tree; superprocesses

摘要

The main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure. The exploration process appears as a simple local time functional of a Levy process with no negative jumps, whose Laplace exponent coincides with the branching mechanism function. This new relation between spectrally positive Levy processes and continuous branching processes provides a unified perspective on both theories. In particular, we derive the adequate formulation of the classical Ray-Knight theorem for such Levy processes. As a consequence of this theorem, we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process.