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November 2017 Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime
David Campos, Alejandro F. Ramírez
Ann. Probab. 45(6B): 4675-4699 (November 2017). DOI: 10.1214/17-AOP1175

Abstract

We consider a random walk in random environment in the low disorder regime on $\mathbb{Z}^{d}$, that is, the probability that the random walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+\varepsilon\xi(x,e)$, where $p(e)$ is deterministic, $\{\{\xi(x,e):\vert e\vert_{1}=1\}:x\in\mathbb{Z}^{d}\}$ are i.i.d. and $\varepsilon>0$ is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in $\varepsilon$ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in $\varepsilon$ for the invariant measure of random perturbations of the simple symmetric random walk in dimensions $d=2$.

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David Campos. Alejandro F. Ramírez. "Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime." Ann. Probab. 45 (6B) 4675 - 4699, November 2017. https://doi.org/10.1214/17-AOP1175

Information

Received: 1 December 2015; Revised: 1 September 2016; Published: November 2017
First available in Project Euclid: 12 December 2017

zbMATH: 1385.60060
MathSciNet: MR3737921
Digital Object Identifier: 10.1214/17-AOP1175

Subjects:
Primary: 60K37, 82C41
Secondary: 82D30

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 6B • November 2017
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