Transient random walk in ℤ 2 with stationary orientations

Françoise Pène

ESAIM: Probability and Statistics (2009)

  • Volume: 13, page 417-436
  • ISSN: 1292-8100

Abstract

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In this paper, we extend a result of Campanino and Pétritis [Markov Process. Relat. Fields 9 (2003) 391–412]. We study a random walk in ℤ 2 with random orientations. We suppose that the orientation of the kth floor is given by ξ k , where ( ξ k ) k ∈ ℤ is a stationary sequence of random variables. Once the environment fixed, the random walk can go either up or down or can stay in the present floor (but moving with respect to its orientation). This model was introduced by Campanino and Pétritis in [Markov Process. Relat. Fields 9 (2003) 391–412] when the ( ξ k ) k ∈ ℤ is a sequence of independent identically distributed random variables. In [Theory Probab. Appl. 52 (2007) 815–826], Guillotin-Plantard and Le Ny extend this result to a situation where the orientations of the floors are independent but chosen with stationary probabilities (not equal to 0 and to 1). In the present paper, we generalize the result of [Markov Process. Relat. Fields 9 (2003) 391–412] to some cases when ( ξ k ) k is stationary. Moreover we extend slightly a result of [Theory Probab. Appl.52 (2007) 815–826].

How to cite

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Pène, Françoise. "Transient random walk in ${\mathbb Z}^2$ with stationary orientations." ESAIM: Probability and Statistics 13 (2009): 417-436. <http://eudml.org/doc/250665>.

@article{Pène2009,
abstract = { In this paper, we extend a result of Campanino and Pétritis [Markov Process. Relat. Fields 9 (2003) 391–412]. We study a random walk in $\{\mathbb Z\}^2$ with random orientations. We suppose that the orientation of the kth floor is given by $\xi_k$, where $(\xi_k)_\{k\in\mathbb Z\}$ is a stationary sequence of random variables. Once the environment fixed, the random walk can go either up or down or can stay in the present floor (but moving with respect to its orientation). This model was introduced by Campanino and Pétritis in [Markov Process. Relat. Fields 9 (2003) 391–412] when the $(\xi_k)_\{k\in\mathbb Z\}$ is a sequence of independent identically distributed random variables. In [Theory Probab. Appl. 52 (2007) 815–826], Guillotin-Plantard and Le Ny extend this result to a situation where the orientations of the floors are independent but chosen with stationary probabilities (not equal to 0 and to 1). In the present paper, we generalize the result of [Markov Process. Relat. Fields 9 (2003) 391–412] to some cases when $(\xi_k)_k$ is stationary. Moreover we extend slightly a result of [Theory Probab. Appl.52 (2007) 815–826]. },
author = {Pène, Françoise},
journal = {ESAIM: Probability and Statistics},
keywords = {Transience; random walk; Markov chain; oriented graphs; stationary orientations.; transience; stationary orientations},
language = {eng},
month = {9},
pages = {417-436},
publisher = {EDP Sciences},
title = {Transient random walk in $\{\mathbb Z\}^2$ with stationary orientations},
url = {http://eudml.org/doc/250665},
volume = {13},
year = {2009},
}

TY - JOUR
AU - Pène, Françoise
TI - Transient random walk in ${\mathbb Z}^2$ with stationary orientations
JO - ESAIM: Probability and Statistics
DA - 2009/9//
PB - EDP Sciences
VL - 13
SP - 417
EP - 436
AB - In this paper, we extend a result of Campanino and Pétritis [Markov Process. Relat. Fields 9 (2003) 391–412]. We study a random walk in ${\mathbb Z}^2$ with random orientations. We suppose that the orientation of the kth floor is given by $\xi_k$, where $(\xi_k)_{k\in\mathbb Z}$ is a stationary sequence of random variables. Once the environment fixed, the random walk can go either up or down or can stay in the present floor (but moving with respect to its orientation). This model was introduced by Campanino and Pétritis in [Markov Process. Relat. Fields 9 (2003) 391–412] when the $(\xi_k)_{k\in\mathbb Z}$ is a sequence of independent identically distributed random variables. In [Theory Probab. Appl. 52 (2007) 815–826], Guillotin-Plantard and Le Ny extend this result to a situation where the orientations of the floors are independent but chosen with stationary probabilities (not equal to 0 and to 1). In the present paper, we generalize the result of [Markov Process. Relat. Fields 9 (2003) 391–412] to some cases when $(\xi_k)_k$ is stationary. Moreover we extend slightly a result of [Theory Probab. Appl.52 (2007) 815–826].
LA - eng
KW - Transience; random walk; Markov chain; oriented graphs; stationary orientations.; transience; stationary orientations
UR - http://eudml.org/doc/250665
ER -

References

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