Brazilian Journal of Probability and Statistics

Asymptotic direction in random walks in random environment revisited

Alexander Drewitz and Alejandro F. Ramírez

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Abstract

Consider a random walk {Xn : n≥0} in an elliptic i.i.d. environment in dimensions d≥2 and call P0 its averaged law starting from 0. Given a direction $l\in\mathbb{S}^{d-1}$, Al={limn→∞Xnl=∞} is called the event that the random walk is transient in the direction l. Recently Simenhaus proved that the following are equivalent: the random walk is transient in the neighborhood of a given direction; P0-a.s. there exists a deterministic asymptotic direction; the random walk is transient in any direction contained in the open half space defined by this asymptotic direction. Here we prove that the following are equivalent: P0(AlAl)=1 in the neighborhood of a given direction; there exists an asymptotic direction ν such that P0(AνAν)=1 and P0-a.s we have $\lim_{n\to\infty}X_{n}/|X_{n}|=\mathbh{1}_{A_{\nu}}\nu-\mathbh{1}_{A_{-\nu}}\nu$; P0(AlAl)=1 if and only if lν≠0. Furthermore, we give a review of some open problems.

Article information

Source
Braz. J. Probab. Stat., Volume 24, Number 2 (2010), 212-225.

Dates
First available in Project Euclid: 20 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1271770269

Digital Object Identifier
doi:10.1214/09-BJPS028

Mathematical Reviews number (MathSciNet)
MR2643564

Zentralblatt MATH identifier
1200.60092

Keywords
Random walk in random environment renewal times asymptotic directions

Citation

Drewitz, Alexander; Ramírez, Alejandro F. Asymptotic direction in random walks in random environment revisited. Braz. J. Probab. Stat. 24 (2010), no. 2, 212--225. doi:10.1214/09-BJPS028. https://projecteuclid.org/euclid.bjps/1271770269


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