A measure-valued diffusion process describing the stepping stone model with infinitely many alleles

Handa, Kenji

doi:10.1016/0304-4149(90)90096-b

Cited by 14 publications

(15 citation statements)

References 4 publications

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“…Definition of the stepping-stone process X Theorem 4.1 below is Theorem 4.1 of [Eva97]. As discussed in Section 1, it is motivated by the characterisation of infinitely-many-types, discrete-sites steppingstone processes via duality with systems of delayed coalescing continuous-time Markov chains (see [DGV95] and [Han90]). Recall that (Ω, F , P) is the probability space on which the processes Z e , Že , ζ e , ξ e , et cetera are defined.…”

Section: The State-space ξ Of the Stepping-stone Processmentioning

confidence: 99%

“…A natural refinement of this two-type diffusion model, considered in [Han90,DGV95], is the corresponding infinitely-many-types model. Here the Fisher-Wright processes at each site are replaced by mutationless Fleming-Viot processes of evolving random probability measures on a suitable uncountable type-space (typically the unit interval [0, 1]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees

Donnelly¹,

Evans²,

Fleischmann³

et al. 2000

Ann. Probab.

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Analogues of stepping-stone models are considered where the sitespace is continuous, the migration process is a general Markov process, and the type-space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum-sites stepping-stone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to 1 2 and, moreover, this space is capacity equivalent to the middle-1 2 Cantor set (and hence also to the Brownian zero set).

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Section: The State-space ξ Of the Stepping-stone Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees

Donnelly¹,

Evans²,

Fleischmann³

et al. 2000

Ann. Probab.

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“…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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“…The general case was studied in [2], where the index set I is either the finite dimensional lattice or the hierarchical group, and the type space is the set of integers. Even though the index set and state space in our model are more general, the proofs are similar to those used in [7] and [2]. For completeness, we sketch a proof below.…”

Section: B(e) Imentioning

confidence: 99%

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

et al. 2011

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A measure-valued diffusion process describing the stepping stone model with infinitely many alleles

Cited by 14 publications

References 4 publications

Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees

Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees

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

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

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