# A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks

@article{Birkner2008AML, title={A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks}, author={Matthias C. F. Birkner and Jochen Blath and Martin Moehle and Matthias Steinruecken and Johanna Tams}, journal={arXiv: Probability}, year={2008} }

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 to the duality known for the classical Fleming-Viot process and Kingman's coalescent, whereis the Dirac measure in 0. We explicitly construct a dual process of the coalescent with simultaneous multi- ple collisions (�-coalescent) with mutation, the �-Fleming-Viot process with muta- tion, and provide…

## 61 Citations

### The reversibility and an SPDE for the generalized Fleming–Viot processes with mutation

- Mathematics
- 2012

### Generalized Fleming-Viot processes with immigration via stochastic flows of partitions

- Mathematics
- 2011

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…

### Pathwise construction of tree-valued Fleming-Viot processes

- Mathematics
- 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…

### The spatial Lambda-Fleming–Viot process: An event-based construction and a lookdown representation

- 2015

We construct a measure-valued equivalent to the spatialΛ-Fleming–Viot process (SLFV) introduced in (Banach Center Publ. 80 (2008) 121–144). In contrast with the construction carried out there, we fix…

### The spatial Lambda-Fleming-Viot process: an event-based construction and a lookdown representation

- Mathematics
- 2012

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…

### Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration

- MathematicsAdvances in Applied Probability
- 2011

Coalescents with multiple collisions (also called Λ-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected…

### Tree-valued resampling dynamics Martingale problems and applications

- Mathematics
- 2008

The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate…

### A representation for exchangeable coalescent trees and generalized tree-valued Fleming-Viot processes

- Mathematics
- 2016

We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We…

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

- Mathematics
- 2013

Using Donnelly and Kurtz's lookdown construction, we prove that the Lambda-Fleming-Viot process with underlying Brownian motion has a compact support at any fixed time provided that the associated…

### The symmetric coalescent and Wright–Fisher models with bottlenecks

- MathematicsThe Annals of Applied Probability
- 2022

We define a new class of $\Xi$-coalescents characterized by a possibly infinite measure over the non negative integers. We call them symmetric coalescents since they are the unique family of…

## References

SHOWING 1-10 OF 46 REFERENCES

### A countable representation of the Fleming-Viot measure-valued diffusion

- Mathematics
- 1996

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…

### Stochastic flows associated to coalescent processes. III. Limit theorems

- Mathematics
- 2005

We prove several limit theorems that relate coalescent processes to continuous-state branching processes. Some of these theorems are stated in terms of the so-called generalized Fleming-Viot…

### Fleming-Viot processes in population genetics

- Mathematics
- 1993

Fleming and Viot [Indiana Univ. Math. J., 28 (1979), pp. 817–843] introduced a class of probability-measure-valued diffusion processes that has attracted the interest of both pure and applied…

### CONVERGENCE TO THE COALESCENT WITH SIMULTANEOUS MULTIPLE MERGERS

- Mathematics
- 2003

The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized by Mohle and Sagitov (2001) in terms of a sequence of measures defined on the…

### Stochastic flows associated to coalescent processes

- Mathematics
- 2003

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…

### Coalescents with multiple collisions

- Mathematics
- 1999

k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's…

### Jump-type Fleming-Viot processes

- MathematicsAdvances in Applied Probability
- 2000

In 1991 Perkins [7] showed that the normalized critical binary branching process is a time inhomogeneous Fleming-Viot process. In the present paper we extend this result to jump-type branching…

### GRAPHICAL REPRESENTATION OF SOME DUALI- TY RELATIONS IN STOCHASTIC POPULATION MO- DELS

- Mathematics
- 2007

We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. [1]) and for the self-duality of Feller’s branching diffusion…

### The general coalescent with asynchronous mergers of ancestral lines

- MathematicsJournal of Applied Probability
- 1999

Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral…