Coagulation-transport equations and the nested coalescents

• 成果类型: Article
• 署名作者: Lambert, Amaury; Schertzer, Emmanuel
• 署名单位: Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Biology (INSB); Universite PSL; College de France; Ecole Normale Superieure (ENS); Institut National de la Sante et de la Recherche Medicale (Inserm)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00914-4
• 发表日期: 2020
• 页码: 77-147
• 关键词: kingman coalescent; nonuniqueness; population; speed; limit

摘要

The nested Kingman coalescent describes the dynamics of particles (called genes) contained in larger components (called species), where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species. We prove that starting from rn species, the empirical distribution of species masses (numbers of genes/n) at time t/n converges as n -> infinity to a solution of the deterministic coagulation-transport equation partial derivative(t)d = partial derivative(x)(psi d) + a(t) (d star d - d), where psi (x) = cx(2), star denotes convolution and a(t) = 1/(t+delta) with delta = 2/r. The most interesting case when delta = 0 corresponds to an infinite initial number of species. This equation describes the evolution of the distribution of species of mass x, where pairs of species can coalesce and each species' mass evolves like (x) over dot = -psi (x). We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when delta = 0. The first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous-state branching processes. The second one is the law of the solution to the following McKean-Vlasov equation dx(t) = -psi (x(t)) dt + v(t) Delta J(t) where J is an inhomogeneous Poisson process with rate 1/(t + delta) and (v(t); t >= 0) is a sequence of independent random variables such that L(v(t)) = L(x(t)). We show that there is a unique solution to this equation and we construct this solution with the help of a marked Brownian coalescent point process. When psi(x) = x(gamma), we show the existence of a self-similar solution for the PDE which relates when gamma = 2 to the speed of coming down from infinity of the nested Kingman coalescent.