Electronic Journal of Probability

Random walks in dynamic random environments and ancestry under local population regulation

Matthias Birkner, Jiří Černý, and Andrej Depperschmidt

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Abstract

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining.

Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high.

Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 38, 43 pp.

Dates
Received: 27 October 2015
Accepted: 23 April 2016
First available in Project Euclid: 26 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464269713

Digital Object Identifier
doi:10.1214/16-EJP4666

Subjects
Primary: 60K37: Processes in random environments 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Random walk dynamical random environment oriented percolation supercritical cluster central limit theorem in random environment logistic branching random walk

Rights
Creative Commons Attribution 4.0 International License.

Citation

Birkner, Matthias; Černý, Jiří; Depperschmidt, Andrej. Random walks in dynamic random environments and ancestry under local population regulation. Electron. J. Probab. 21 (2016), paper no. 38, 43 pp. doi:10.1214/16-EJP4666. https://projecteuclid.org/euclid.ejp/1464269713.


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