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

Evans, Steven N.

doi:10.1016/s0246-0203(97)80095-7

Cited by 28 publications

(49 citation statements)

References 12 publications

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“…Note that (ii) corroborates a remark at the beginning of §5 in [Eva97]. In Evans' construction, all the genealogical processes used as duals are made up of independent Hunt processes that coalesce instantaneously upon meeting.…”

Section: Duality Between the Slfv And Its Genealogiessupporting

confidence: 75%

“…Now, Theorem 4.1 in [Eva97] guarantees that there exists a unique Ξ-valued Markov process starting from (the equivalence class of) 1 H (x) and dual to A ∞ through the relations (8). Let us call this process ρ (2) .…”

Section: Duality Between the Slfv And Its Genealogiesmentioning

confidence: 99%

“…The state-space Ξ of the SLFV is then defined as the quotient spaceΞ/ ∼ of equivalence classes of ∼. Ξ is endowed with a natural topology described in §3 of [Eva97]. As mentioned earlier, it is possible to check that this topology is equivalent to the topology of vague convergence of the measures ρ(x 1 , dκ 1 ) .…”

Section: Convergence and Duality 31 State-space And Form Of Convergencementioning

confidence: 99%

“…The same kind of behaviour, as well as the convergence of the forwards-in-time process to a field of correlated Bernoulli random variables, was already observed by Evans in the case where the genealogical process of his continuous sites stepping-stone model is a system of one-dimensional independent α-stable motions coalescing instantly upon meeting. See §5 in [Eva97] for a full description of his results. However, the underlying mechanisms are quite different here.…”

Section: Heavy-tailed Casementioning

confidence: 99%

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Large scale behaviour of the spatial $\varLambda$-Fleming–Viot process

Berestycki¹,

Etheridge²,

Véber³

2013

Ann. Inst. H. Poincaré Probab. Statist.

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We consider the spatial Λ-Fleming-Viot process model ([BEV10]) for frequencies of genetic types in a population living in R d , in the special case in which there are just two types of individuals, labelled 0 and 1. At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type 0. We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the dynamics of the process are driven by purely 'local' events and one incorporating large-scale extinction recolonisation events. We choose the frequency of these events in such a way that, under a suitable rescaling of space and time, the ancestry of a single individual in the population converges to a symmetric stable process of index α ∈ (1, 2] (with α = 2 corresponding to Brownian motion). We consider the behaviour of the process of allele frequencies under the same space and time rescaling. For α = 2, and d ≥ 2 it converges to a deterministic limit. In all other cases the limit is random and we identify it as the indicator function of a random set. In particular, there is no local coexistence of types in the limit. We characterise the set in terms of a dual process of coalescing symmetric stable processes, which is of interest in its own right. The complex geometry of the random set is illustrated through simulations.Onétudie le processus Λ-Fleming-Viot spatial ([BEV10]) modélisant les fréquences locales de types génétiques dans une populationévoluant dans R d . On considère le cas particulier où il n'y a que deux types possibles, notés 0 et 1. Initialement, tous les individus présents dans le demi-espace des points dont la première coordonnée est négative sont de type 1, tandis que les individus présents dans le demi-espace complémentaire sont de type 0. On s'intéresse au comportement des fréquences locales sur deséchelles de temps et d'espace très grandes. On considère deux cas : dans le premier, l'évolution du processus est due uniquementà deś evénements 'locaux'; dans le second, on incorpore desévénements d'extinction et recolonisation de grande ampleur. On choisit la fréquence de cesévénements de sorte qu'après une renormalisation spatiale et temporelle appropriée, la lignée ancestrale d'un individu de la population converge vers un processus α-stable symétrique, d'indice α ∈ (1, 2] (où α = 2 correspond au mouvement brownien). Onétudie l'évolution du processus des fréquences alléliques aux mêmeséchelles spatio-temporelles. Lorsque α = 2 et d ≥ 2, celui-ci converge vers un processus déterministe. Dans tous les autres cas, le processus limite est aléatoire et on l'identifie comme la fonction indicatrice d'un ensemble aléatoireévoluant au cours du temps. En particulier, les deux types ne coexistent pasà la limite. On caractérise chaque ensemble en termes d'un processus dual constitué de mouvements stables symétriques coalescents ayant un intérêt en eux-mêmes. La géométrie complexe des ensembles limites est illustrée par des simulations.

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Section: Duality Between the Slfv And Its Genealogiessupporting

confidence: 75%

Section: Duality Between the Slfv And Its Genealogiesmentioning

confidence: 99%

Section: Convergence and Duality 31 State-space And Form Of Convergencementioning

confidence: 99%

Section: Heavy-tailed Casementioning

confidence: 99%

See 2 more Smart Citations

Large scale behaviour of the spatial $\varLambda$-Fleming–Viot process

Berestycki¹,

Etheridge²,

Véber³

2013

Ann. Inst. H. Poincaré Probab. Statist.

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“…In Barton et al (2010), through a powerful result of Evans (1997), it is shown that existence of the SLFV can be deduced from existence of the dual. In that paper, by assuming that 4) one guarantees that started from any finite number of individuals, the jump rate in the dual is finite, and so Definition 2.4 gives rise to a well-defined process.…”

Section: )mentioning

confidence: 99%

SuperBrownian motion and the spatial Lambda-Fleming-Viot process

Chetwynd-Diggle¹,

Etheridge²

2018

Electron. J. Probab.

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It is well known that the dynamics of a subpopulation of individuals of a rare type in a Wright-Fisher diffusion can be approximated by a Feller branching process. Here we establish an analogue of that result for a spatially distributed population whose dynamics are described by a spatial Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is then approximated by a superBrownian motion. This result mirrors Cox et al. (2000), where it is shown that when suitably rescaled, sparse voter models converge to superBrownian motion. We also prove the somewhat more surprising result, that by choosing the dynamics of the SLFV appropriately we can recover superBrownian motion with stable branching in an analogous way. This is a spatial analogue of (a special case of) results of Bertoin and Le Gall (2006), who show that the generalised Fleming-Viot process that is dual to the beta-coalescent, when suitably rescaled, converges to a continuous state branching process with stable branching mechanism.

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Coalescing systems of non-Brownian particles

Evans

Morris

Sen

2012

Probab. Theory Relat. Fields

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A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions S. N. Evans supported in part by NSF grants DMS-0405778 and DMS-0907630 and B. Morris supported in part by NSF grant DMS-0707144.

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

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

Cited by 28 publications

References 12 publications

Large scale behaviour of the spatial $\varLambda$-Fleming–Viot process

Large scale behaviour of the spatial $\varLambda$-Fleming–Viot process

SuperBrownian motion and the spatial Lambda-Fleming-Viot process

Coalescing systems of non-Brownian particles

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