GENERAL SELECTION MODELS: BERNSTEIN DUALITY AND MINIMAL ANCESTRAL STRUCTURES

• 成果类型: Article
• 署名作者: Cordero, Fernando; Hummel, Sebastian; Schertzer, Emmanuel
• 署名单位: University of Bielefeld; Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1683
• 发表日期: 2022
• 页码: 1499-1556
• 关键词: block counting process; coalescent processes; common ancestor; stochastic-equations; fixation line; lambda; diffusion; genealogy; FLOWS; graph

摘要

Lambda-Wright-Fisher processes provide a robust framework to describe the type-frequency evolution of an infinite neutral population. We add a polynomial drift to the corresponding stochastic differential equation to incorporate frequency-dependent selection. A decomposition of the drift allows us to approximate the solution of the stochastic differential equation by a sequence of Moran models. The genealogical structure underlying the Moran model leads in the large population limit to a generalisation of the ancestral selection graph of Krone and Neuhauser. Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and criteria for the accessibility of the boundaries; this extends a recent result by Gonzalez Casanova and Spano. An intriguing feature of the construction is that the same forward process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones.