Electronic Journal of Probability

Random walk in random environment in a two-dimensional stratified medium with orientations

Alexis Devulder and Françoise Pène

Full-text: Open access

Abstract

We consider a model of random walk in ${\mathbb Z}^2$ with (fixed or random) orientation of the horizontal lines (layers) and with non constant iid probability to stay on these lines. We prove the transience of the walk for any fixed orientations under general hypotheses. This contrasts with the model of Campanino and Petritis, in which probabilities to stay on these lines are all equal. We also establish a result of convergence in distribution for this walk with suitable normalizations under more precise assumptions. In particular, our model proves to be, in many cases, even more superdiffusive than the random walks introduced by Campanino and Petritis.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 18, 23 pp.

Dates
Accepted: 29 January 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064243

Digital Object Identifier
doi:10.1214/EJP.v18-2459

Mathematical Reviews number (MathSciNet)
MR3035746

Zentralblatt MATH identifier
1283.60058

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G52: Stable processes 60K37: Processes in random environments

Keywords
random walk on randomly oriented lattices random walk in random environment random walk in random scenery functional limit theorem transience

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

Citation

Devulder, Alexis; Pène, Françoise. Random walk in random environment in a two-dimensional stratified medium with orientations. Electron. J. Probab. 18 (2013), paper no. 18, 23 pp. doi:10.1214/EJP.v18-2459. https://projecteuclid.org/euclid.ejp/1465064243


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References

  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
  • Bouchaud, J.-P.; Georges, A.; Koplik, J.; Provata, A. and Redner, S.: Superdiffusion in Random Velocity Fields. phPhys. Rev. letters 64 (21), (1990), 2503–2506.
  • Campanino, M.; Petritis, D. Random walks on randomly oriented lattices. Markov Process. Related Fields 9 (2003), no. 3, 391–412.
  • Campanino, M. and Petritis, D.: Type transition of simple random walks on randomly directed regular lattices. ARXIV1204.5297.
  • Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise; Schapira, Bruno. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Ann. Probab. 39 (2011), no. 6, 2079–2118.
  • Dombry, C.; Guillotin-Plantard, N. Discrete approximation of a stable self-similar stationary increments process. Bernoulli 15 (2009), no. 1, 195–222.
  • Gantert, Nina; König, Wolfgang; Shi, Zhan. Annealed deviations of random walk in random scenery. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007), no. 1, 47–76.
  • Guillotin-Plantard, N.; Le Ny, A. Transient random walks on 2D-oriented lattices. Teor. Veroyatn. Primen. 52 (2007), no. 4, 815–826; translation in Theory Probab. Appl. 52 (2008), no. 4, 699–711
  • Guillotin-Plantard, Nadine; Le Ny, Arnaud. A functional limit theorem for a 2D-random walk with dependent marginals. Electron. Commun. Probab. 13 (2008), 337–351.
  • Heyde, C. C. On the asymptotic behavior of random walks on an anisotropic lattice. J. Statist. Phys. 27 (1982), no. 4, 721–730.
  • Heyde, C. C.; Westcott, M.; Williams, E. R. The asymptotic behavior of a random walk on a dual-medium lattice. J. Statist. Phys. 28 (1982), no. 2, 375–380.
  • Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman. Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp.
  • Jain, Naresh C.; Pruitt, William E. Asymptotic behavior of the local time of a recurrent random walk. Ann. Probab. 12 (1984), no. 1, 64–85.
  • Kesten, H.; Spitzer, F. A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 5–25.
  • de Loynes, B.: Random walk on a directed graph and Martin boundary, ARXIV1203.3306.
  • Lukacs, Eugene. Characteristic functions. Second edition, revised and enlarged. Hafner Publishing Co., New York, 1970. x+350 pp.
  • Matheron, G. and de Marsily, G.: Is transport in porous media always diffusive? A counterexample. phWater Resources Res. 16 (5), (1980), 901–917.
  • Nagaev, S. V. On large deviations of a self-normalized sum. (Russian) Teor. Veroyatn. Primen. 49 (2004), no. 4, 794–802; translation in Theory Probab. Appl. 49 (2005), no. 4, 704–713
  • Pène, Françoise. Transient random walk in $\Bbb Z^ 2$ with stationary orientations. ESAIM Probab. Stat. 13 (2009), 417–436.