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

TL;DR

This paper studies random walks in evolving environments modeled by Markov processes, establishing laws of large numbers and central limit theorems, with applications to ancestral lineages in spatial population models.

Contribution

It introduces a regeneration-based approach to analyze random walks in dynamic environments generated by Markov processes, applicable to population genetics models.

Findings

Law of large numbers for the walk under certain conditions

Central limit theorem for the walk in typical environments

Application to logistic branching random walks at high density

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.