Residual Diffusivity in Elephant Random Walk Models with Stops

Abstract

We study the enhanced diffusivity in the so called elephant random walk model with stops (ERWS) by including symmetric random walk steps at small probability $\epsilon$. At any $\epsilon > 0$, the large time behavior transitions from sub-diffusive at $\epsilon = 0$ to diffusive in a wedge shaped parameter regime where the diffusivity is strictly above that in the un-perturbed ERWS model in the $\epsilon \downarrow 0$ limit. The perturbed ERWS model is shown to be solvable with the first two moments and their asymptotics calculated exactly in both one and two space dimensions. The model provides a discrete analytical setting of the residual diffusion phenomenon known for the passive scalar transport in chaotic flows (e.g. generated by time periodic cellular flows and statistically sub-diffusive) as molecular diffusivity tends to zero.