## Electronic Journal of Probability

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

#### Abstract

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, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming–Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent.

In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the Fleming–Viot genealogies at fixed time t arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than ε apart in the limit ε → 0. Second, we answer two open questions about the sample paths of the tree-valued Fleming–Viot process. We show that for all t > 0 almost surely the marked metric measure space Xt has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming–Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov’s formula giving absolute continuity.

#### Article information

Source
Electron. J. Probab. Volume 18 (2013), paper no. 84, 47 pp.

Dates
Accepted: 19 September 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064309

Digital Object Identifier
doi:10.1214/EJP.v18-2514

Mathematical Reviews number (MathSciNet)
MR3109623

Zentralblatt MATH identifier
1284.60166

Rights

#### Citation

Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Path-properties of the tree-valued Fleming–Viot process. Electron. J. Probab. 18 (2013), paper no. 84, 47 pp. doi:10.1214/EJP.v18-2514. https://projecteuclid.org/euclid.ejp/1465064309.

#### References

• Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3–48.
• Berestycki, Julien; Berestycki, NathanaÃ«l. Kingman's coalescent and Brownian motion. ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), 239–259.
• Delmas, Jean-François; Dhersin, Jean-Stéphane; Siri-Jegousse, Arno. On the two oldest families for the Wright-Fisher process. Electron. J. Probab. 15 (2010), no. 26, 776–800.
• Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Marked metric measure spaces. Electron. Commun. Probab. 16 (2011), 174–188.
• Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Tree-valued Fleming-Viot dynamics with mutation and selection. Ann. Appl. Probab. 22 (2012), no. 6, 2560–2615.
• Donnelly, Peter; Kurtz, Thomas G. A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24 (1996), no. 2, 698–742.
• Donnelly, Peter; Kurtz, Thomas G. Genealogical processes for Fleming-Viot models with selection and recombination. Ann. Appl. Probab. 9 (1999), no. 4, 1091–1148.
• Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
• Ethier, S. N.; Kurtz, Thomas G. Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 (1993), no. 2, 345–386.
• Evans, Steven N.; Ralph, Peter L. Dynamics of the time to the most recent common ancestor in a large branching population. Ann. Appl. Probab. 20 (2010), no. 1, 1–25.
• Stochastic models. Proceedings of the International Conference (ICSM) held in honour of Professor Donald A. Dawson at Carleton University, Ottawa, ON, June 10â€“13, 1998. Edited by Luis G. Gorostiza and B. Gail Ivanoff. CMS Conference Proceedings, 26. Published by the American Mathematical Society, Providence, RI; for the Canadian Mathematical Society, Ottawa, ON, 2000. xxxviii+450 pp. ISBN: 0-8218-1063-4
• Feller, William. An introduction to probability theory and its applications. Vol. II. John Wiley & Sons, Inc., New York-London-Sydney 1966 xviii+636 pp.
• Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 (2009), no. 1-2, 285–322.
• Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields 155 (2013), no. 3-4, 789–838.
• Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
• Kingman, J. F. C. The coalescent. Stochastic Process. Appl. 13 (1982), no. 3, 235–248.
• Kingman, J. F. C. Exchangeability and the evolution of large populations. Exchangeability in probability and statistics (Rome, 1981), pp. 97–112, North-Holland, Amsterdam-New York, 1982.
• Pfaffelhuber, P.; Wakolbinger, A. The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 (2006), no. 12, 1836–1859.
• Pfaffelhuber, P.; Wakolbinger, A.; Weisshaupt, H. The tree length of an evolving coalescent. Probab. Theory Related Fields 151 (2011), no. 3-4, 529–557.
• R. Pyke. The continuum limit of critical random graphs. Journal of the Royal Statistical Society. Series B., 27:395–449, 1965.
• Schweinsberg, Jason. Dynamics of the evolving Bolthausen-Sznitman coalecent. [Dynamics of the evolving Bolthausen-Sznitman coalescent] Electron. J. Probab. 17 (2012), no. 91, 50 pp.
• Tavaré, Simon. Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol. 26 (1984), no. 2, 119–164.