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

Matthias Birkner, Jiří Černý, Andrej Depperschmidt

Published 2015-05-11Version 1

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 the spatial embedding of an ancestral lineage 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, thus partly settling a question from Depperschmidt (2008) on the behaviour of their ancestral lines.