A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks

Abstract

Let $\Lambda$ be a finite measure on the unit interval. A $\Lambda$-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions ($\Lambda$-coalescent) in analogy to the duality known for the classical Fleming Viot process and Kingman's coalescent, where $\Lambda$ is the Dirac measure in 0. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions ($\Xi$-coalescent) with mutation, the $\Xi$-Fleming-Viot process with mutation, and provide a representation based on the empirical measure of an exchangeable particle system along the lines of Donnelly and Kurtz (1999). We establish pathwise convergence of the approximating systems to the limiting $\Xi$-Fleming-Viot process with mutation. An alternative construction of the semigroup based on the Hille-Yosida theorem is provided and various types of duality of the processes are discussed. In the last part of the paper a population is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $\Xi$-Fleming-Viot processes naturally arise as limiting models.

35 pages, 2 figures

Cited by in corpus (21)

• The site frequency spectrum for general coalescents
• Linkage disequilibrium under recurrent bottlenecks
• Multi-locus data distinguishes between population growth and multiple merger coalescents
• Recent progress in coalescent theory
• Computational inference beyond Kingman's coalescent
• From flows of Lambda Fleming-Viot processes to lookdown processes via flows of partitions
• Genealogies and inference for populations with highly skewed offspring distributions
• On the notion(s) of duality for Markov processes
• Generalized Fleming-Viot processes with immigration via stochastic flows of partitions
• Approximate filtering via discrete dual processes
• Consistency of Bayesian nonparametric inference for discretely observed jump diffusions
• Tree-valued resampling dynamics: Martingale Problems and applications
• On the speed of coming down from infinity for $\X$-coalescent processes
• Dual process in the two-parameter Poisson-Dirichlet diffusion
• Some support properties for a class of Lambda-Fleming-Viot processes
• Weak atomic convergence of finite voter models toward Fleming-Viot processes
• Gene genealogies in haploid populations evolving according to sweepstakes reproduction
• An ancestral recombination graph for diploid populations with skewed offspring distribution
• The compact support property for the $\Lambda$-Fleming-Viot process with underlying Brownian motion
• On Exceptional Times for generalized Fleming-Viot Processes with Mutations
• The reversibility and an SPDE for the generalized Fleming-Viot Processes with mutation