Quasi-invariance and reversibility in the Fleming–Viot process

Handa, Kenji

doi:10.1007/s004400100178

Cited by 14 publications

(20 citation statements)

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“…Some particular examples of our applications have already been studied in the literature: necessary and sufficient conditions for two subordinators to have equivalent laws are given in Sato [10, p. 217-218]; we have already mentioned the result of Tsilevich, Vershik and Yor in [12] concerning local equivalence of γ and ( t 0 a s dγ s , t ≥ 0), for any deterministic measurable function a : R + → R + with a and 1/a bounded, with explicit Radon-Nikodym density; in the case of the Dirichlet process, two distinct quasi-invariance properties have been studied by Handa in [5] and von Renesse and Sturm in [7]. We refer to the remarks after each result in Sections 3 and 4, where these results are recalled in detail.…”

Section: A Look At the Bibliographymentioning

confidence: 96%

“…We note that our approach allows us to treat the previously mentioned results of Vershik, Tsilevich and Yor [12,13], together with Handa's [5] and the recent work by von Renesse and Sturm [7] on Dirichlet processes, within a unified framework.…”

Section: Introductionmentioning

confidence: 95%

“…[12,3,5]). Quasi-invariance properties of the associated probability measure on path or measure space with respect to canonical transformations often play a central role.…”

Section: Introductionmentioning

confidence: 96%

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Quasi-invariance properties of a class of subordinators

Renesse¹,

Yor²,

Zambotti³

2008

Stochastic Processes and their Applications

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We study absolute-continuity relationships for a class of stochastic processes, including the gamma and the Dirichlet processes. We prove that the laws of a general class of non-linear transformations of such processes are locally equivalent to the law of the original process and we compute explicitly the associated Radon-Nikodym densities. This work unifies and generalizes to random non-linear transformations several previous quasi-invariance results for gamma and Dirichlet processes.

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Section: A Look At the Bibliographymentioning

confidence: 96%

Section: Introductionmentioning

confidence: 95%

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Quasi-invariance properties of a class of subordinators

Renesse¹,

Yor²,

Zambotti³

2008

Stochastic Processes and their Applications

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“…Despite the interchange of x and y, it is still possible to interpret the alternative form for the transition density in terms of a dual process running backwards in time (Donnelly and Tavaré, 1987;Etheridge and Griffiths, 2009). However, the Wright-Fisher diffusion with recombination is not reversible (Handa, 2002).…”

Section: A Transition Function Expansionmentioning

confidence: 99%

A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning

Griffiths

Jenkins

Lessard

2016

Theoretical Population Biology

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Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the Wright-Fisher diffusion for genetic drift and its genealogical counterpart, the coalescent. There have been a number of articles extending this relationship to include other evolutionary processes such as mutation and selection, but little has been explored for models also incorporating crossover recombination. Here, we derive from first principles a new genealogical process which is dual to a Wright-Fisher diffusion model of drift, mutation, and recombination. The process is reminiscent of the ancestral recombination graph, a widely-used multilocus genealogical model, but here ancestral lineages are typed and transition rates are regarded as being conditioned on an observed configuration at the leaves of the genealogy. Our approach is based on expressing a putative duality relationship between two models via their infinitesimal generators, and then seeking an appropriate test function to ensure the validity of the duality equation. This approach is quite general, and we use it to find dualities for several important variants, including both a discrete L-locus model of a gene and a continuous model in which mutation and recombination events are scattered along the gene according to * Corresponding author. Address: Department of Statistics, University of Warwick, Coventry CV4 7AL, United Kingdom. E-mail: p.jenkins@warwick.ac.uk continuous distributions. As an application of our results, we derive a series expansion for the transition function of the diffusion. Finally, we study in further detail the case in which mutation is absent. Then the dual process describes the dispersal of ancestral genetic material across the ancestors of a sample. The stationary distribution of this process is of particular interest; we show how duality relates this distribution to haplotype fixation probabilities. We develop an efficient method for computing such probabilities in multilocus models.

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“…We then show that X has a unique invariant measure if the mutation process allows a unique invariant measure.The reversibility of a population genetic model is an important issue for statistical inference. 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].…”

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confidence: 99%

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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Quasi-invariance and reversibility in the Fleming–Viot process

Cited by 14 publications

References 13 publications

Quasi-invariance properties of a class of subordinators

Quasi-invariance properties of a class of subordinators

A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning

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

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