Reversibility of Interacting Fleming–Viot Processes with Mutation, Selection, and Recombination

Feng, Shui; Schmuland, Byron; Vaillancourt, Jean; Zhou, Xiaowen

doi:10.4153/cjm-2010-071-1

Cited by 4 publications

(3 citation statements)

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“…It was also shown in Li et al (1999) that the classical Fleming-Viot process with mutation operator of uniform type is reversible. Feng et al (2011) proved the irreversibility for an interacting Fleming-Viot processes with mutation, selection and recombination. The reversibility for the generalized Ξ-Fleming-Viot process was investigated in Li et al (2013), where the authors proved that the Ξ-Fleming-Viot process is not reversible except for the degenerate case of classical Fleming-Viot process with mutation operator of uniform type.…”

Section: Introductionmentioning

confidence: 98%

“…Kermany et al (2008) derived expressions for joint stationary moments of a two-island diffusion model and represented the sampling formula in terms of the joint moments. More recent work on stepping stone model can be found in Feng et al (2011) where the stepping stone model is characterized as a system of interacting Fleming-Viot processes and its irreversibility was discussed.…”

Section: Introductionmentioning

confidence: 99%

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Generalized stepping stone model with $Ξ$-resampling mechanism

Liu,

Zhou

2020

Preprint

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In this paper we formulate a generalized stepping stone model with Ξresampling mechanism to describe the evolution of relative frequencies for different types of alleles in a population with migration between two colonies. For a Ξ-coalescent and a jump type mutation generator A, such a probability-measure-valued Markov process is dual to the (Ξ, A)-coalescent process with geographical labels and migration. The existence of the generalized stepping stone model is directly established from a moment duality by verifying a multidimensional Hausdorff moment problem, and its probability law is also uniquely determined by the moment duality. Further, we characterize the stationary distribution for this model and show that the model is not reversible when the mutation operator is of uniform type.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Generalized stepping stone model with $Ξ$-resampling mechanism

Liu,

Zhou

2020

Preprint

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“…The reversibility for the classical Fleming-Viot process has been investigated in Li et al [18] using Dirichlet forms and in Handa [12] and Schmuland and Sun [24] via cocycle identity. The reversibility for an interacting classical Fleming-Viot process is studied in Feng et al [11]. We also consider the reversibility for (Ξ, A)-Fleming-Viot process in this paper.…”

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confidence: 98%

The reversibility and an SPDE for the generalized Fleming–Viot processes with mutation

Liu

Xiong

et al. 2013

Stochastic Processes and their Applications

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The (Ξ, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the Ξ-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.[23]) is a coalescent with possible simultaneous multiple collisions. It is then interesting to know whether there exists a generalized Fleming-Viot type probability-measure-valued process whose dual is a function-valued process evolving in the same way as the classical Fleming-Viot dual but with the Kingman's coalescent replaced by the Ξ-coalescent.Such a generalized Fleming-Viot process was first considered by Donnelly and Kurtz [9] and Hiraba [13]. When the spatial motion of the particle is negated, namely, the mutation is 0, it has also been studied by Bertoin and Le Gall ([2], [3], [4], [5]) and Birkner et al [7]. In particular, a special form of such process is constructed in [4] using the weak solution flow of a stochastic equation driven by a Poisson random measure. Generalized Fleming-Viot processes with parent independent jump mutation operators are constructed by Dawson and Li [8] as strong solutions of stochastic equations driven by time-space white noises and Poisson random measures. The classical Fleming-Viot process with Laplacian mutation operator is characterized by Xiong [25] as the strong solution of an SPDE driven by a time-space white noise. The common feature of the approaches of [4], [8] and [25] is to consider the processes of distributions of the measurevalued processes instead of their density processes. In fact, the processes studied in [4] and [8] are usually not absolutely continuous. Similar stochastic equations for Dawson-Watanabe superprocesses have also been studied in [4], [5], [8] and [25].This problem is also studied in the recent work of Birkner et al [6]. When the mutation generator A is the generator for a pure jump Markov process, two constructions of the (Ξ, A)-Fleming-Viot process are found in [6]. One construction is based on modification of the lookdown scheme of [9] applied to exchangeable particle systems for the classical Fleming-Viot process. The (Ξ, A)-Fleming-Viot process arises as the pathwise almost sure limit of the empirical measure for the exchangeable particle system. The other construction is based on the Hille-Yosida theorem. The resulted process gives an example of probability-measure-valued superprocess of jump diffusion type.In this paper, we further study the existence and various properties of this generalized Fleming-Viot process. We first formulate a well-posed martingale problem and show that the (Ξ, A)-Fleming-Viot process X with general mutation generator is the unique solution to such a martingale problem. We then show that X has...

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Generalized Stepping Stone Model with Ξ-resampling Mechanism

Liu

Zhou²

2022

Acta. Math. Sin.-English Ser.

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Reversibility of Interacting Fleming–Viot Processes with Mutation, Selection, and Recombination

Cited by 4 publications

References 15 publications

Generalized stepping stone model with $Ξ$-resampling mechanism

Generalized stepping stone model with $Ξ$-resampling mechanism

The reversibility and an SPDE for the generalized Fleming–Viot processes with mutation

Generalized Stepping Stone Model with Ξ-resampling Mechanism

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