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2012 Scaling limits of recurrent excited random walks on integers
Dmitry Dolgopyat, Elena Kosygina
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Electron. Commun. Probab. 17: 1-14 (2012). DOI: 10.1214/ECP.v17-2213


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.


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Dmitry Dolgopyat. Elena Kosygina. "Scaling limits of recurrent excited random walks on integers." Electron. Commun. Probab. 17 1 - 14, 2012.


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

zbMATH: 1252.60098
MathSciNet: MR2965748
Digital Object Identifier: 10.1214/ECP.v17-2213

Primary: 60K37
Secondary: 60F17 , 60G50

Keywords: branching process , Cookie walk , excited random walk , perturbed Brownian motion , random environment


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