Electronic Communications in Probability

The genealogy of an exactly solvable Ornstein–Uhlenbeck type branching process with selection

Aser Cortines and Bastien Mallein

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Abstract

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 \to \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.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 98, 13 pp.

Dates
Received: 6 February 2018
Accepted: 19 November 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1545102494

Digital Object Identifier
doi:10.1214/18-ECP197

Mathematical Reviews number (MathSciNet)
MR3896836

Zentralblatt MATH identifier
07023484

Subjects
Primary: S0J80 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92D15: Problems related to evolution

Keywords
branching random walk selection beta coalescent Poisson point process

Rights
Creative Commons Attribution 4.0 International License.

Citation

Cortines, Aser; Mallein, Bastien. The genealogy of an exactly solvable Ornstein–Uhlenbeck type branching process with selection. Electron. Commun. Probab. 23 (2018), paper no. 98, 13 pp. doi:10.1214/18-ECP197. https://projecteuclid.org/euclid.ecp/1545102494


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