Pathwise construction of tree-valued Fleming-Viot processes

@article{Gufler2014PathwiseCO,
  title={Pathwise construction of tree-valued Fleming-Viot processes},
  author={Stephan Gufler},
  journal={arXiv: Probability},
  year={2014}
}
In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple reproduction events. We construct families of probability measures on the lookdown space and on an extension of it that allows to include the case with dust. From this construction, we read off the tree-valued $\Xi$-Fleming-Viot processes and deduce path properties. For instance, these processes… 

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References

SHOWING 1-10 OF 42 REFERENCES
The spatial Lambda-Fleming-Viot process: an event-based construction and a lookdown representation
We construct a measure-valued equivalent to the spatial Lambda-Fleming-Viot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of
A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks
Letbe a finite measure on the unit interval. A �-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions (�-coalescent) in analogy
Invariance principles for tree-valued Cannings chains
We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under
Stochastic flows associated to coalescent processes
Abstract. We study a class of stochastic flows connected to the coalescent processes that have been studied recently by Möhle, Pitman, Sagitov and Schweinsberg in connection with asymptotic models
Generalized Fleming-Viot processes with immigration via stochastic flows of partitions
The generalized Fleming-Viot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key
Trickle-down processes and their boundaries
TLDR
A framework that encompasses Markov chains is introduced, and their asymptotic behavior is characterized by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail $\sigma$-fields.
From flows of Λ-Fleming-Viot processes to lookdown processes via flows of partitions
The goal of this paper is to unify the lookdown representation and the stochastic flow of bridges, which are two approaches to construct the Λ-Fleming-Viot process along with its genealogy. First we
Marked metric measure spaces
A marked metric measure space (mmm-space) is a triple $(X,r,μ)$, where $(X,r)$ is a complete and separable metric space and $μ$ is a probability measure on $X \times I$ for some Polish space $I$ of
A countable representation of the Fleming-Viot measure-valued diffusion
The Fleming-Viot measure-valued diffusion arises as the infinite population limit of various discrete genetic models with general type space. The paper gives a countable construction of the process
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure
...
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