### The Λ-Fleming-Viot process and a connection with Wright-Fisher diffusion

Robert C. Griffiths

#### Abstract

The d-dimensional Λ-Fleming-Viot generator acting on functions g(x), with x being a vector of d allele frequencies, can be written as a Wright-Fisher generator acting on functions g with a modified random linear argument of x induced by partitioning occurring in the Λ-Fleming-Viot process. The eigenvalues and right polynomial eigenvectors are easy to see from this representation. The two-dimensional process, which has a one-dimensional generator, is considered in detail. A nonlinear equation is found for the Green's function. In a model with genic selection a proof is given that there is a critical selection value such that if the selection coefficient is greater than or equal to the critical value then fixation, when the boundary 1 is hit, has probability 1 beginning from any nonzero frequency. This is an analytic proof different from the proofs of Der, Epstein and Plotkin (2011) and Foucart (2013). An application in the infinitely-many-alleles Λ-Fleming-Viot process is finding an interesting identity for the frequency spectrum of alleles that is based on size biasing. The moment dual process in the Fleming-Viot process is the usual Λ-coalescent tree back in time. The Wright-Fisher representation using a different set of polynomials gn(x) as test functions produces a dual death process which has a similarity to the Kingman coalescent and decreases by units of one. The eigenvalues of the process are analogous to the Jacobi polynomials when expressed in terms of gn(x), playing the role of xn. Under the stationary distribution when there is mutation, E[gn(X)] is analogous to the nth moment in a beta distribution. There is a d-dimensional version gn(X), and even an intriguing Ewens' sampling formula analogy when d → ∞.

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 4 (2014), 1009-1035.

Dates
First available in Project Euclid: 12 December 2014

https://projecteuclid.org/euclid.aap/1418396241

Digital Object Identifier
doi:10.1239/aap/1418396241

Mathematical Reviews number (MathSciNet)
MR3290427

Zentralblatt MATH identifier
1305.60038

Subjects
Primary: 60G99: None of the above, but in this section
Secondary: 92D15: Problems related to evolution

#### Citation

Griffiths, Robert C. The Λ-Fleming-Viot process and a connection with Wright-Fisher diffusion. Adv. in Appl. Probab. 46 (2014), no. 4, 1009--1035. doi:10.1239/aap/1418396241. https://projecteuclid.org/euclid.aap/1418396241

#### References

• Bah, B. and Pardoux, E. (2013). Lambda-lookdown model with selection. Preprint. Available at http://uk.arxiv.org/ abs/1303.1953.
• Berestycki, N. (2009). Recent Progress in Coalescent Theory (Ensaios Matemáticos 16). Sociedade Brasileira de Matemática, Rio de Janeiro.
• Berestycki, J., Berestycki, N. and Limic, V. (2014a). A small-time coupling between $\Lambda$-coalescents and branching processes. Ann. Appl. Prob. 24, 449–475.
• Berestycki, J., Berestycki, N. and Limic, V. (2014b). Asymptotic sampling formulae for $\Lambda$-coalescents. Ann. Inst. H. Poincaré Prob. Statist.50, 715–731
• Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Prob. Theory Relat. Fields 126, 261–288.
• Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50, 147–181.
• Birkner, M. and Blath, J. (2009). Measure-valued diffusions, general coalescents and population genetic inference. In Trends in Stochastic Analysis, Cambridge University Press, pp. 329–363.
• Birkner, M. et al. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Prob. 10, 303–325.
• Cox, D. R. (1970). Renewal Theory. Methuen, London.
• Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.
• Der, R., Epstein, C. L. and Plotkin, J. B. (2011). Generalized population models and the nature of genetic drift. Theoret. Pop. Biol. 80, 80–99.
• Der, R., Epstein, C. and Plotkin, J. B. (2012). Dynamics of neutral and selected alleles when the offspring distribution is skewed. Genetics 191, 1331–1344.
• Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming–Viot measure-valued diffusion. Ann. Prob. 24, 698–742.
• Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Prob. 27, 166–205.
• Eldon, B. and Wakeley, J. (2006). Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172, 2621–2633.
• Etheridge, A. (2011). Some Mathematical Models from Population Genetics. Springer, Heidelberg.
• Etheridge, A. M., Griffiths, R. C. and Taylor, J. E. (2010). A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit. Theoret. Pop. Biol. 78, 77–92.
• Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87–112.
• Ewens, W. J. (2004). Mathematical Population Genetics. I. Theoretical Introduction, 2nd edn. Springer, New York.
• Foucart, C. (2013). The impact of selection in the $\Lambda$–Wright–Fisher model. Electron. Commun. Prob. 18, no. 72. (Erratum: 19 (2014), no. 15.)
• Griffiths, R. C. (1980). Lines of descent in the diffusion approximation of neutral Wright–Fisher models. Theoret. Pop. Biol. 17, 37–50.
• Handa, K. (2014). Stationary distributions for a class of generalized Fleming–Viot processes. Ann. Prob. 42, 1257–1284.
• Ismail, M. E. H. (2005). Classical and Quantum Orthogonal Polynomials in One variable (Encyclopedia Math. Appl. 98). Cambridge University Press.
• Kimura, M. (1964). Diffusion models in population genetics. J. Appl. Prob. 1, 177–232.
• Lessard, S. (2010). Recurrence equations for the probability distribution of sample configurations in exact population genetics models. J. Appl. Prob. 47, 732–751.
• Möhle, M. (2006). On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12, 35–53.
• Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547–1562.
• Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.
• Pitman, J. (2006). Combinatorial Stochastic Processes. Springer, Berlin.
• Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.
• Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Commun. Prob. 5, 1–11.
• Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton–Watson processes. Stoch. Process. Appl. 106, 107–139.
• Tavaré, S. (1984). Line-of-descent and genealogical processes, and their application in population genetics models. Theoret. Pop. Biol. 26, 119–164. \endharvreferences