Electronic Journal of Probability

SuperBrownian motion and the spatial Lambda-Fleming-Viot process

Jonathan A. Chetwynd-Diggle and Alison M. Etheridge

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Abstract

It is well known that the dynamics of a subpopulation of individuals of a rare type in a Wright-Fisher diffusion can be approximated by a Feller branching process. Here we establish an analogue of that result for a spatially distributed population whose dynamics are described by a spatial Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is then approximated by a superBrownian motion. This result mirrors [10], where it is shown that when suitably rescaled, sparse voter models converge to superBrownian motion. We also prove the somewhat more surprising result, that by choosing the dynamics of the SLFV appropriately we can recover superBrownian motion with stable branching in an analogous way. This is a spatial analogue of (a special case of) results of [6], who show that the generalised Fleming-Viot process that is dual to the beta-coalescent, when suitably rescaled, converges to a continuous state branching process with stable branching mechanism.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 71, 36 pp.

Dates
Received: 29 May 2017
Accepted: 25 June 2018
First available in Project Euclid: 26 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1532570599

Digital Object Identifier
doi:10.1214/18-EJP191

Mathematical Reviews number (MathSciNet)
MR3835477

Zentralblatt MATH identifier
06924683

Subjects
Primary: 60F05: Central limit and other weak theorems 60G57: Random measures 60J25: Continuous-time Markov processes on general state spaces 60J68: Superprocesses 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G51: Processes with independent increments; Lévy processes 60G55: Point processes 60J75: Jump processes 92D10f

Keywords
spatial Lambda-Fleming-Viot model superprocess stable branching process scaling limits

Rights
Creative Commons Attribution 4.0 International License.

Citation

Chetwynd-Diggle, Jonathan A.; Etheridge, Alison M. SuperBrownian motion and the spatial Lambda-Fleming-Viot process. Electron. J. Probab. 23 (2018), paper no. 71, 36 pp. doi:10.1214/18-EJP191. https://projecteuclid.org/euclid.ejp/1532570599


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