# Tree-valued Fleming–Viot dynamics with mutation and selection

@article{Depperschmidt2012TreevaluedFD, title={Tree-valued Fleming–Viot dynamics with mutation and selection}, author={Andrej Depperschmidt and Andreas Greven and Peter Pfaffelhuber}, journal={Annals of Applied Probability}, year={2012}, volume={22}, pages={2560-2615} }

The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces…

## Figures from this paper

## 57 Citations

### Path-properties of the tree-valued Fleming-Viot process

- Mathematics
- 2012

We consider the tree-valued Fleming–Viot process, (Xt )t≥0 , with mutation and selection. This process models the stochastic evolution of the genealogies and (allelic) types under resampling,…

### An Ergodic Theorem for Fleming-Viot Models in Random Environments

- Mathematics
- 2017

The Fleming-Viot (FV) process is a measure-valued diffusion that models the evolution of type frequencies in a countable population which evolves under resampling (genetic drift), mutation, and…

### Duality for spatially interacting Fleming-Viot processes with mutation and selection

- Mathematics
- 2011

Consider a system X = ((x�(t)),� 2 N)t�0 of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space (P(I))N, where I is…

### 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…

### Continuum Space Limit of the Genealogies of Interacting Fleming-Viot Processes on $\Z$

- Mathematics
- 2015

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…

### Tree-valued Feller diffusion

- Mathematics
- 2019

We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion model. In contrast to the approach via labeled trees in the continuum random tree world,…

### Invariance principles for tree-valued Cannings chains

- Mathematics
- 2016

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…

### The historical Moran model

- Mathematics
- 2015

The limit theorem has two applications and it is obtained that the conditioned genealogical distance of two individuals given the types of the two individuals is distributed as a certain stopping time of a further functional of the backward process which is a new approach towards a proof that genealogyical distances are stochastically smaller under selection.

### Measure representation of evolving genealogies.

- Mathematics
- 2019

We study evolving genealogies, i.e. processes that take values in the space of (marked) ultra-metric measure spaces and satisfy some sort of "consistency" condition. This condition is based on the…

### 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…

## References

SHOWING 1-10 OF 73 REFERENCES

### 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…

### Genealogical processes for Fleming-Viot models with selection and recombination

- Mathematics
- 1999

Infinite population genetic models with general type space incorporating mutation, selection and recombination are considered. The Fleming– Viot measure-valued diffusion is represented in terms of a…

### Duality for spatially interacting Fleming-Viot processes with mutation and selection

- Mathematics
- 2011

Consider a system X = ((x�(t)),� 2 N)t�0 of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space (P(I))N, where I is…

### 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…

### 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…

### A Fleming—Viot process with unbounded selection

- Mathematics
- 2008

Tachida (1991) proposed a discrete-time model o f nearly neutral m utation in w hich the selection coefficient of a new mutant has a fixed normal distribution with mean 0. The usual diffusion…

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

- Mathematics
- 1995

In this paper of infinite systems of interacting measure-valued diffusions each with state space ¿^([O, 1]), the set of probability measures on [0, 1], is constructed and analysed (Fleming-Viot…

### 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…

### 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…

### Spatial Fleming-Viot Models with Selection and Mutation

- Mathematics
- 2013

Introduction.- Emergence and fixation in the F-W model with two types.- Formulation of the multitype and multiscale model.- Formulation of the main results in the general case.- A Basic Tool: Dual…