Equilibria and Quasi-Equilibria for Infinite Collections of Interacting Fleming-Viot Processes

Dawson, Donald A.; Greven, Andreas; Vaillancourt, Jean

doi:10.2307/2154827

Cited by 27 publications

(44 citation statements)

References 26 publications

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“…Cerrai and Clément [6], [7] proved that for S the simplex and the hypercube, respectively, H contains a large subset of H. This work represents very important progress on the difficult weak uniqueness issue in the anisotropic case. The stochastic part of the renormalization program in the anisotropic case is also largely open, with partial progress in Dawson, Greven and Vaillancourt [13] for S the simplex.…”

Section: Definementioning

confidence: 99%

Renormalization of Hierarchically Interacting Isotropic Diffusions

Hollander

Swart

1998

Journal of Statistical Physics

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Systems of hierarchically interacting diffusions allow for a rigorous renormalization analysis. By bringing into play the powerful machinery of stochastic analysis, it is possible to obtain a complete classification of the large space-time behavior of these systems into universality classes. The present paper outlines a general renormalization program that is being pursued since ten years and describes four examples where this program has been successfully carried through. The systems under consideration model the evolution of multi-type populations subject to migration and resampling.

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Section: Definementioning

confidence: 99%

Renormalization of Hierarchically Interacting Isotropic Diffusions

Hollander

Swart

1998

Journal of Statistical Physics

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“…The form of the duality is that multivariate moments for the system of diffusions can be represented as certain expectations for the delayed coalescing Markov chains. Shiga's observation has been particularly useful in studying the phenomenon of cluster formation in such models (see [7] or [3] for recent bibliographies covering papers in this area). Shiga [13] also showed that 341 COALESCING MARKOV LABELLED PARTITIONS the natural stochastic partial differential equation analogue of such a system of interacting Fisher-Wright diffusions is dual in a similar way to a system of delayed coalescing Markov processes.…”

Section: Introductionmentioning

confidence: 99%

“…The above two-type genetics models have infinitely-many-types counterparts, and duality for these has been investigated in [9] and [3].…”

Section: Introductionmentioning

confidence: 99%

Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types

Evans

1997

Annales de l'Institut Henri Poincare (B) Probability and Statis

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“…We will use several known facts on measure-valued Fleming-Viot diffusions. For that we refer to [Daw93, Chapter 5] for the non-spatial case and to [DGV95] for the spatial case.…”

Section: Proof Of Theorems 112 and 117mentioning

confidence: 99%

Continuum space limit of the genealogies of interacting Fleming-Viot processes on $\mathbb{Z}$

Greven¹,

Sun²,

Winter³

2016

Electron. J. Probab.

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We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on Z. We construct and analyze the genealogical structure of the population in this genealogy-valued Fleming-Viot process as a marked metric measure space, with each individual carrying its spatial location as a mark. We then show that its time evolution converges to that of the genealogy of a continuum-sites stepping stone model on R, if space and time are scaled diffusively. We construct the genealogies of the continuum-sites stepping stone model as functionals of the Brownian web, and furthermore, we show that its evolution solves a martingale problem. The generator for the continuum-sites stepping stone model has a singular feature: at each time, the resampling of genealogies only affects a set of individuals of measure 0. Along the way, we prove some negative correlation inequalities for coalescing Brownian motions, as well as extend the theory of marked metric measure spaces (developed recently by Depperschmidt, Greven and Pfaffelhuber [DGP12]) from the case of probability measures to measures that are finite on bounded sets.AMS 2010 subject classification: 60K35, 60J65, 60J70, 92D25. 1(1.6) (X n , r n , ψ k · µ n ) =⇒ n→∞ (X, r, ψ k · µ) in the Gromov-weak topology for each k ∈ N.When V = R d , we may choose ψ k to be infinitely differentiable. Remark 1.3 (Dependence on o and (ψ k ) k∈N ). Note that the V -marked Gromov-weak # topology does not depend on the choice o ∈ V and the sequence (ψ k ) k∈N , as long as ψ k has bounded support and A k := {v : ψ k (v) = 1} increases to V as k → ∞.Remark 1.4 (M V as a subspace of (M V f ) N ). Let M V f denote the space of (equivalent classes of) V -mmm spaces with finite measures, equipped with the V -marked Gromovweak topology as introduced in [DGP11, Def. 2.4]. Then it is a well-known fact that each element (X, r, µ) ∈ M V can be identified with a sequence ((X, r, ψ 1 · µ), (X, r, ψ 2 · µ), . . .) in the product space (M V f ) N , equipped with the product topology. This identification allows us to easily deduce many properties of M V from properties of M V f that were established in [DGP11]. In particular, we can metrize the V -marked Gromov-weak # topology on M V by introducing a metric (which can be called V -marked Gromov-Prohorov # metric) (1.7)where d M GP is the marked Gromov-Prohorov metric on M V f , which was introduced in [DGP11, Def. 3.1] and metrizes the marked Gromov-weak topology.The proof of the following result is in Appendix A.Theorem 1.5 (Polish space). The space M V , equipped with the V -marked Gromov-weak # topology, is a Polish space.

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Equilibria and Quasi-Equilibria for Infinite Collections of Interacting Fleming-Viot Processes

Cited by 27 publications

References 26 publications

Renormalization of Hierarchically Interacting Isotropic Diffusions

Renormalization of Hierarchically Interacting Isotropic Diffusions

Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types

Continuum space limit of the genealogies of interacting Fleming-Viot processes on $\mathbb{Z}$

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