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

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.