Communications in Mathematical Sciences

Volume 16 (2018)

Number 7

Residual diffusivity in elephant random walk models with stops

Pages: 2033 – 2045



Jiancheng Lyu (Department of Mathematics, University of California at Irvine)

Jack Xin (Department of Mathematics, University of California at Irvine)

Yifeng Yu (Department of Mathematics, University of California at Irvine)


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 $\epsilon \gt 0$, 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 unperturbed 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.


elephant random walk with stops, sub-diffusion, moment analysis, residual diffusivity

2010 Mathematics Subject Classification

58J37, 60G50, 60H30

The work was partly supported by NSF grants DMS-1211179 (JX), DMS-1522383 (JX), DMS-0901460 (YY), and CAREER Award DMS-1151919 (YY).

Received 15 October 2017

Accepted 22 July 2018

Published 7 March 2019