The genealogy of continuous-state branching processes with immigration

• 成果类型: Article
• 署名作者: Lambert, A
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400100155
• 发表日期: 2002
• 页码: 42-70
• 关键词: Levy processes; excursions; infimum

摘要

Recent works by J.F. Le Gall and Y. Le Jan [ 15] have extended the genealogical structure of Galton-Watson processes to continous-state branching processes (CB). We are here interested in processes with immigration (CBI). The height process H which contains all the information about this genealogical structure is defined as a simple local time functional of a strong Markov process X*, called the genealogy-coding process (GCP). We first show its existence using Ito's synthesis theorem. We then give a pathwise construction of X* based on a Levy process X with no negative jumps that does not drift to +infinity and whose Laplace exponent coincides with the branching mechanism, and an independent subordinator Y whose Laplace exponent coincides with the immigration mechanism. We conclude the construction with proving that the local time process of H is a CBI-process. As an application, we derive the analogue of the classical Ray-Knight-Williams theorem for a general Levy process with no negative jump.