THE RELATIVE FREQUENCY BETWEEN TWO CONTINUOUS-STATE BRANCHING PROCESSES WITH IMMIGRATION AND THEIR GENEALOGY

• 成果类型: Article
• 署名作者: Emilia Caballero, Maria; Gonzalez Casanov, Adrian; Perez, Jose-Luis
• 署名单位: Universidad Nacional Autonoma de Mexico
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1991
• 发表日期: 2024
• 页码: 1271-1318
• 关键词: within-generation variance; stochastic-equations; coalescent processes; natural-selection; offspring number; REPRESENTATION

摘要

When two (possibly different in distribution) continuous-state branching processes with immigration are present, we study the relative frequency of one of them when the total mass is forced to be constant at a dense set of times. This leads to a SDE whose unique strong solution will be the definition of a Lambda-asymmetric frequency process (Lambda-AFP). We prove that it is a Feller process and we calculate a large population limit when the total mass tends to infinity. This allows us to study the fluctuations of the process around its deterministic limit. Furthermore, we find conditions for the Lambda-AFP to have a moment dual. The dual can be interpreted in terms of selection, (coordinated) mutation, pairwise branching (efficiency), coalescence, and a novel component that comes from the asymmetry between the reproduction mechanisms. In the particular case of a pair of equally distributed continuous-state branching processes the associated Lambda-AFP will be the dual of a Lambda-coalescent. The map that sends each continuous-state branching process to its associated Lambda-coalescent (according to the former procedure) is a homeomorphism between metric spaces.