×

zbMATH — the first resource for mathematics

The genealogy of an exactly solvable Ornstein-Uhlenbeck type branching process with selection. (English) Zbl 1406.60130
Summary: We study the genealogy of an exactly solvable population model with \(N\) particles on the real line, which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around \(a\) times its current position, where \(a>0\) is a parameter of the model. Then, the \(N\) rightmost newborn children are selected to form the next generation. We show that the genealogy of the process converges toward a Beta coalescent as \(N \rightarrow \infty \). The process we consider can be seen as a toy model version of a continuous-time branching process with selection, in which particles move according to independent Ornstein–Uhlenbeck processes. The parameter \(a\) is akin to the pulling strength of the Ornstein–Uhlenbeck process.
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
92D15 Problems related to evolution
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Radosław Adamczak and Piotr Miłoś, \(U\)-statistics of Ornstein-Uhlenbeck branching particle system, J. Theoret. Probab. 27 (2014), no. 4, 1071–1111. · Zbl 1311.60093
[2] Radosław Adamczak and Piotr Miłoś, CLT for Ornstein-Uhlenbeck branching particle system, Electron. J. Probab. 20 (2015), no. 42, 35.
[3] Louis-Pierre Arguin, Anton Bovier, and Nicola Kistler, The extremal process of branching Brownian motion, Probab. Theory Related Fields 157 (2013), no. 3-4, 535–574. · Zbl 1286.60045
[4] E. Aïdékon, J. Berestycki, É. Brunet, and Z. Shi, Branching Brownian motion seen from its tip, Probab. Theory Related Fields 157 (2013), no. 1-2, 405–451.
[5] Jean Bérard and Jean-Baptiste Gouéré, Brunet-Derrida behavior of branching-selection particle systems on the line, Comm. Math. Phys. 298 (2010), no. 2, 323–342. · Zbl 1247.60124
[6] Jean Bérard and Pascal Maillard, The limiting process of \(N\)-particle branching random walk with polynomial tails, Electron. J. Probab. 19 (2014), no. 22, 17. · Zbl 1295.60105
[7] Julien Berestycki, Nathanaël Berestycki, and Jason Schweinsberg, The genealogy of branching Brownian motion with absorption, Ann. Probab. 41 (2013), no. 2, 527–618. · Zbl 1304.60088
[8] Nathanaël Berestycki, Recent progress in coalescent theory, Ensaios Matemáticos [Mathematical Surveys], vol. 16, Sociedade Brasileira de Matemática, Rio de Janeiro, 2009.
[9] E. Bolthausen and A.-S. Sznitman, On Ruelle’s probability cascades and an abstract cavity method, Comm. Math. Phys. 197 (1998), no. 2, 247–276. · Zbl 0927.60071
[10] É. Brunet, B. Derrida, A. H. Mueller, and S. Munier, Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization, Phys. Rev. E (3) 76 (2007), no. 4, 041104, 20.
[11] Eric Brunet and Bernard Derrida, Shift in the velocity of a front due to a cutoff, Phys. Rev. E (3) 56 (1997), no. 3, part A, 2597–2604.
[12] Francis Comets and Aser Cortines, Finite-size corrections to the speed of a branching-selection process, Braz. J. Probab. Stat. 31 (2017), no. 3, 476–501. · Zbl 1377.82027
[13] Aser Cortines, The genealogy of a solvable population model under selection with dynamics related to directed polymers, Bernoulli 22 (2016), no. 4, 2209–2236. · Zbl 1344.60092
[14] Aser Cortines and Bastien Mallein, A \(N\)-branching random walk with random selection, ALEA Lat. Am. J. Probab. Math. Stat. 14 (2017), no. 1, 117–137. · Zbl 1358.60088
[15] Olivier Couronné and Lucas Gerin, A branching-selection process related to censored Galton-Walton processes, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 1, 84–94. · Zbl 1286.60085
[16] A. De Masi, P.A. Ferrari, E. Presutti, and N. Soprano-Loto, Non-local branching Brownians with annihilation and free boundary problems, Preprint, 2017.
[17] Rick Durrett and Daniel Remenik, Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations, Ann. Probab. 39 (2011), no. 6, 2043–2078. · Zbl 1243.60066
[18] R. A. Fisher, The genetical theory of natural selection, variorum ed., Oxford University Press, Oxford, 1999, Revised reprint of the 1930 original, Edited, with a foreword and notes, by J. H. Bennett. · JFM 56.1106.13
[19] J. F. C. Kingman, On the genealogy of large populations, J. Appl. Probab. (1982), no. Special Vol. 19A, 27–43, Essays in statistical science. · Zbl 0516.92011
[20] B. Mallein, \(N\)-branching random walk with \(α \)-stable spine, Teor. Veroyatn. Primen. 62 (2017), no. 2, 365–392, Reprinted in Theory Probab. Appl. 62 (2018), no. 2, 295–318.
[21] Bastien Mallein, Branching random walk with selection at critical rate, Bernoulli 23 (2017), no. 3, 1784–1821. · Zbl 1392.60070
[22] Michel Pain, Velocity of the \(L\)-branching Brownian motion, Electron. J. Probab. 21 (2016), Paper No. 28, 28. · Zbl 1336.60168
[23] K. Pearson and A. Lee, On the laws of inheritance in man: I. Inheritance of physical characters, Biometrika 2 (1903), 357–462.
[24] Jim Pitman, Coalescents with multiple collisions, Ann. Probab. 27 (1999), no. 4, 1870–1902. · Zbl 0963.60079
[25] R.V. Rohlfs, P. Harrigan, and R. Nielsen, Modeling gene expression evolution with an extended Ornstein–Uhlenbeck process accounting for Within-species variation, Mol. Biol. Evol. 31 (2014), no. 1, 201–211.
[26] Serik Sagitov, The general coalescent with asynchronous mergers of ancestral lines, J. Appl. Probab. 36 (1999), no. 4, 1116–1125. · Zbl 0962.92026
[27] Jason Schweinsberg, Coalescent processes obtained from supercritical Galton-Watson processes, Stochastic Process. Appl. 106 (2003), no. 1, 107–139. · Zbl 1075.60571
[28] Q. Shi, A growth-fragmentation model related to Ornstein-Uhlenbeck type processes, Preprint, 2017.
[29] S. Wright, Evolution in Mendelian populations, Genetics 16 (1931), 97–159.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.