Electronic Communications in Probability

Scaling limits of recurrent excited random walks on integers

Dmitry Dolgopyat and Elena Kosygina

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We describe scaling limits of recurrent excited random walks (ERWs) on $\mathbb{Z}$ in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, $\delta$, belongs to the interval $[-1,1]$. We show that if $|\delta|<1$ then the diffusively scaled ERW under the averaged measure converges to a $(\delta,-\delta)$-perturbed Brownian motion. In the boundary case, $|\delta|=1$, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens to be a constant multiple of the running maximum of the standard Brownian motion, a transient process.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 35, 14 pp.

Accepted: 9 August 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 60F17: Functional limit theorems; invariance principles 60G50: Sums of independent random variables; random walks

excited random walk cookie walk branching process random environment perturbed Brownian motion

This work is licensed under a Creative Commons Attribution 3.0 License.


Dolgopyat, Dmitry; Kosygina, Elena. Scaling limits of recurrent excited random walks on integers. Electron. Commun. Probab. 17 (2012), paper no. 35, 14 pp. doi:10.1214/ECP.v17-2213. https://projecteuclid.org/euclid.ecp/1465263168

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