Electronic Journal of Probability

Continuum space limit of the genealogies of interacting Fleming-Viot processes on $\mathbb{Z}$

Andreas Greven, Rongfeng Sun, and Anita Winter

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We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on $\mathbb{Z}$. We construct and analyze the genealogical structure of the population in this genealogy-valued Fleming-Viot process as a marked metric measure space, with each individual carrying its spatial location as a mark. We then show that its time evolution converges to that of the genealogy of a continuum-sites stepping stone model on $\mathbb{R}$, if space and time are scaled diffusively. We construct the genealogies of the continuum-sites stepping stone model as functionals of the Brownian web, and furthermore, we show that its evolution solves a martingale problem. The generator for the continuum-sites stepping stone model has a singular feature: at each time, the resampling of genealogies only affects a set of individuals of measure $0$. Along the way, we prove some negative correlation inequalities for coalescing Brownian motions, as well as extend the theory of marked metric measure spaces (developed recently by Depperschmidt, Greven and Pfaffelhuber [DGP11]) from the case of probability measures to measures that are finite on bounded sets.

Article information

Electron. J. Probab. Volume 21, Number (2016), paper no. 58, 64 pp.

Received: 28 August 2015
Accepted: 12 September 2016
First available in Project Euclid: 30 September 2016

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Digital Object Identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J65: Brownian motion [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D25: Population dynamics (general)

Brownian web continuum-sites stepping stone model evolving genealogies interacting Fleming-Viot process marked metric measure space martingale problems negative correlation inequalities spatial continuum limit

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Greven, Andreas; Sun, Rongfeng; Winter, Anita. Continuum space limit of the genealogies of interacting Fleming-Viot processes on $\mathbb{Z}$. Electron. J. Probab. 21 (2016), paper no. 58, 64 pp. doi:10.1214/16-EJP4514. http://projecteuclid.org/euclid.ejp/1475266506.

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