Wandering Random Measures in the Fleming-Viot Model

Dawson, Donald A.; Hochberg, Kenneth J.

doi:10.1214/aop/1176993767

Cited by 76 publications

(58 citation statements)

References 24 publications

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“…Note that moment formulas for Fleming-Viot type process are classical (see e.g. Dawson and Hochberg (1982) or (Dawson, 1993, Section 2.8)); we show here that they are actually available much more broadly.…”

Section: Introductionmentioning

confidence: 70%

Probability measure-valued polynomial diffusions

Cuchiero¹,

Larsson²,

Svaluto-Ferro³

2019

Electron. J. Probab.

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We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming-Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero et al. (2012); Filipović and Larsson (2016) is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

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Section: Introductionmentioning

confidence: 70%

Probability measure-valued polynomial diffusions

Cuchiero¹,

Larsson²,

Svaluto-Ferro³

2019

Electron. J. Probab.

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“…In previous contexts, such a model has been used to model quantitative trait evolution, called a continuum-of-alleles model [Kimura, 1965]. Earlier approaches to incorporate genetic drift into a continuum-of-alleles model using the theory of measure-valued diffusions [Fleming and Viot, 1979, Ethier and Kurtz, 1987] have made significant advances in understanding the neutral dynamics of such processes [Dawson and Hochberg, 1982, Ethier and Griffiths, 1993, Ethier and Kurtz, 1993, Donnelly and Kurtz, 1996]. However, incorporating selection greatly increases the difficulty of obtaining analytical results (but see Donnelly and Kurtz [1999], Dawson and Feng [2001]).…”

Section: Discussionmentioning

confidence: 99%

A path integral formulation of the Wright–Fisher process with genic selection

Schraiber

2014

Theoretical Population Biology

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The Wright-Fisher process with selection is an important tool in population genetics theory. Traditional analysis of this process relies on the diffusion approximation. The diffusion approximation is usually studied in a partial differential equations framework. In this paper, I introduce a path integral formalism to study the Wright-Fisher process with selection and use that formalism to obtain a simple perturbation series to approximate the transition density. The perturbation series can be understood in terms of Feynman diagrams, which have a simple probabilistic interpretation in terms of selective events. The perturbation series proves to be an accurate approximation of the transition density for weak selection and is shown to be arbitrarily accurate for any selection coefficient.

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“…The Λ-Fleming-Viot process does not have a compact support if the associated Λ-coalescent does not come down from infinity. Liu and Zhou [20] recently extended the results in [9] to a class of Λ-Fleming-Viot processes whose associated Λ-coalescents come down from infinity. We are not aware of any results on the modulus of continuity for Fleming-Viot support processes although the modulus of continuity for superBrownian motion support had been first recovered by Dawson et al [11] more than twenty years ago and further studied in Dawson and Vinogradov [12] and in Dawson el al.…”

Section: Introductionmentioning

confidence: 92%

“…Consequently, there are only a few results available for Fleming-Viot support processes so far. The earliest work on the compact support property for classical Fleming-Viot processes is due to Dawson and Hochberg [9]. It was shown in [9] that at any fixed time T > 0 the classical Fleming-Viot process with underlying Brownian motion has a compact support with Hausdorff dimension not greater than two.…”

Section: Introductionmentioning

confidence: 99%

“…The earliest work on the compact support property for classical Fleming-Viot processes is due to Dawson and Hochberg [9]. It was shown in [9] that at any fixed time T > 0 the classical Fleming-Viot process with underlying Brownian motion has a compact support with Hausdorff dimension not greater than two. Using non-standard techniques Reimers [23] improved the above result by showing that the carrying dimension of the support is at most two simultaneously for all positive times.…”

Section: Introductionmentioning

confidence: 99%

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Some support properties for a class of ${\varLambda}$-Fleming–Viot processes

Liu¹,

Zhou²

2015

Ann. Inst. H. Poincaré Probab. Statist.

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For a class of Λ-Fleming-Viot processes with underlying Brownian motion whose associated Λ-coalescents come down from infinity, we prove a one-sided modulus of continuity result for their ancestry processes recovered from the lookdown construction of Donnelly and Kurtz. As applications, we first show that such a Λ-Fleming-Viot support process has one-sided modulus of continuity (with modulus function C t log (1/t)) at any fixed time. We also show that the support is compact simultaneously at all positive times, and given the initial compactness, its range is uniformly compact over any finite time interval. In addition, under a mild condition on the Λ-coalescence rates, we find a uniform upper bound on Hausdorff dimension of the support and an upper bound on Hausdorff dimension of the range.

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Wandering Random Measures in the Fleming-Viot Model

Cited by 76 publications

References 24 publications

Probability measure-valued polynomial diffusions

Probability measure-valued polynomial diffusions

A path integral formulation of the Wright–Fisher process with genic selection

Some support properties for a class of ${\varLambda}$-Fleming–Viot processes

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