Branching processes in lévy processes: The exploration process

Jean Francois Le Gall, Yves Le Jan

Research output: Contribution to journalArticlepeer-review

Abstract

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 Lévy process with no negative jumps, whose Laplace exponent coincides with the branching mechanism function. This new relation between spectrally positive Lévy 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 Lévy 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.

Original languageEnglish (US)
Pages (from-to)213-252
Number of pages40
JournalAnnals of Probability
Volume26
Issue number1
DOIs
StatePublished - Jan 1998

Keywords

  • Branching processes
  • Exploration process
  • Genealogy
  • Jump processes
  • Lévy processes
  • Local time
  • Random tree

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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