Gene flow across geographical barriers - scaling limits of random walks with obstacles

TL;DR

This paper investigates the scaling limits of a class of random walks with barriers, revealing a process that alternates between reflected Brownian motion and jumps at exponential times, with applications in population genetics.

Contribution

It introduces a novel scaling limit for random walks with barriers, characterizing a process that alternates between reflected Brownian motion and jumps at exponential times, with explicit formulas and martingale problem characterization.

Findings

The scaling limit behaves like reflected Brownian motion until local time reaches an exponential level.

The process alternates between reflected Brownian motion on either side of the barrier.

Explicit transition density formula for the limiting process.

Abstract

In this paper, we study the scaling limit of a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. If the probability of crossing the barrier scales as $ 1/\sqrt{n} $ as we rescale space by $ \sqrt{n} $ and time by $ n $, we obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at the origin reaches an independent exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at the origin reaches another exponential level, and so on. We give a martingale problem characterisation of this process as well as another construction and an explicit formula for its transition density. This result has applications in the field of population genetics where such a random walk is used to trace the position of one's ancestor in the past in the presence of a barrier to gene flow.