## Electronic Journal of Probability

### Sub-ballistic random walk in Dirichlet environment

Élodie Bouchet

#### Abstract

We consider random walks in Dirichlet environment (RWDE) on $\mathbb{Z} ^d$, for $d \geq 3$, in the sub-ballistic case. We associate to any parameter $(\alpha_1, \dots, \alpha _{2d})$ of the Dirichlet law a time-change to accelerate the walk. We prove that the continuous-time accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to characterize directional transience for the initial RWDE. It solves as a corollary the problem of Kalikow's $0-1$ law in the Dirichlet case in any dimension. Furthermore, we find the polynomial order of the magnitude of the original walk's displacement.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 58, 25 pp.

Dates
Accepted: 25 May 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064283

Digital Object Identifier
doi:10.1214/EJP.v18-2109

Mathematical Reviews number (MathSciNet)
MR3068389

Zentralblatt MATH identifier
1296.60267

Rights

#### Citation

Bouchet, Élodie. Sub-ballistic random walk in Dirichlet environment. Electron. J. Probab. 18 (2013), paper no. 58, 25 pp. doi:10.1214/EJP.v18-2109. https://projecteuclid.org/euclid.ejp/1465064283

#### References

• Atkinson, Giles. Recurrence of co-cycles and random walks. J. London Math. Soc. (2) 13 (1976), no. 3, 486–488.
• Bolthausen, Erwin; Sznitman, Alain-Sol. Ten lectures on random media. DMV Seminar, 32. BirkhÃ¤user Verlag, Basel, 2002. vi+116 pp. ISBN: 3-7643-6703-2
• Durrett, Richard. Probability: theory and examples. Second edition. Duxbury Press, Belmont, CA, 1996. xiii+503 pp. ISBN: 0-534-24318-5
• Enriquez, Nathanaël; Sabot, Christophe. Edge oriented reinforced random walks and RWRE. C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 941–946.
• Enriquez, Nathanaël; Sabot, Christophe. Random walks in a Dirichlet environment. Electron. J. Probab. 11 (2006), no. 31, 802–817 (electronic).
• Kalikow, Steven A. Generalized random walk in a random environment. Ann. Probab. 9 (1981), no. 5, 753–768.
• Keane, M. S.; Rolles, S. W. W. Tubular recurrence. Acta Math. Hungar. 97 (2002), no. 3, 207–221.
• Kesten, H.; Kozlov, M. V.; Spitzer, F. A limit law for random walk in a random environment. Compositio Math. 30 (1975), 145–168.
• Krengel, Ulrich. Ergodic theorems. With a supplement by Antoine Brunel. de Gruyter Studies in Mathematics, 6. Walter de Gruyter & Co., Berlin, 1985. viii+357 pp. ISBN: 3-11-008478-3
• Lyons, Russell and Peres, Yuval: Probabilities on trees and networks. Cambridge University Press. In preparation, available at http://mypage.iu.edu/string rdlyons/.
• Pemantle, Robin. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988), no. 3, 1229–1241.
• Sabot, Christophe. Random walks in random Dirichlet environment are transient in dimension $d\geq 3$. Probab. Theory Related Fields 151 (2011), no. 1-2, 297–317.
• Sabot, Christophe: Random Dirichlet environment viewed from the particle in dimension d geq 3. To appear in Annals of Probability. arxiv:1007.2565
• Sabot, Christophe; Tournier, Laurent. Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Ann. Inst. Henri PoincarÃ© Probab. Stat. 47 (2011), no. 1, 1–8.
• Sinaï, Ya. G. The limit behavior of a one-dimensional random walk in a random environment. (Russian) Teor. Veroyatnost. i Primenen. 27 (1982), no. 2, 247–258.
• Solomon, Fred. Random walks in a random environment. Ann. Probability 3 (1975), 1–31.
• Sznitman, Alain-Sol; Zerner, Martin. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999), no. 4, 1851–1869.
• Tournier, Laurent. Integrability of exit times and ballisticity for random walks in Dirichlet environment. Electron. J. Probab. 14 (2009), no. 16, 431–451.
• Tournier, Laurent: Asymptotic direction of random walks in Dirichlet environment. Preprint. arxiv:1205.6199
• Zeitouni, Ofer. Random walks in random environment. Lectures on probability theory and statistics, 189–312, Lecture Notes in Math., 1837, Springer, Berlin, 2004.
• Zerner, Martin P. W.; Merkl, Franz. A zero-one law for planar random walks in random environment. Ann. Probab. 29 (2001), no. 4, 1716–1732.