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# Communications in Mathematical Sciences

## Volume 16 (2018)

### Number 7

### Residual diffusivity in elephant random walk models with stops

Pages: 2033 – 2045

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n7.a12

#### Authors

#### 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 $\epsilon \gt 0$, 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.

#### Keywords

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