Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Averaged large deviations for random walk in a random environment

Atilla Yilmaz

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Abstract

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤd with d≥1, and gives a variational formula for the corresponding rate function Ia. Under Sznitman’s transience condition (T), we show that Ia is strictly convex and analytic on a non-empty open set $\mathcal{A}$, and that the true velocity of the particle is an element (resp. in the boundary) of $\mathcal{A}$ when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan’s variational formula at any velocity in $\mathcal{A}$.

Résumé

Dans son article de 2003, Varadhan démontre un principe de grandes déviations pour la loi moyennée de la vitesse d’une particule suivant une marche aléatoire au plus proche voisin dans un environnement i.i.d. elliptique sur ℤd avec d≥1, et donne une formule variationnelle pour la fonction de taux correspondante Ia. Sous la condition (T) de transience de Sznitman, nous montrons que Ia est strictement convexe et analytique dans un ouvert non vide ${\mathcal{A}}$, et que la vraie vitesse de la particule est un élément de ${\mathcal{A}}$ (resp. un élément de la frontière de ${\mathcal{A}}$>) quand la marche est “non-nichée” (resp. nichée). Nous identifions alors l’unique minimisant de la formule variationnelle de Varadhan pour toute velocité de ${\mathcal{A}}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 3 (2010), 853-868.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1281100401

Digital Object Identifier
doi:10.1214/09-AIHP332

Mathematical Reviews number (MathSciNet)
MR2682269

Zentralblatt MATH identifier
1201.60098

Subjects
Primary: 60K37: Processes in random environments 60F10: Large deviations 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
Disordered media Rare events Rate function Regeneration times

Citation

Yilmaz, Atilla. Averaged large deviations for random walk in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 3, 853--868. doi:10.1214/09-AIHP332. https://projecteuclid.org/euclid.aihp/1281100401


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References

  • [1] N. Berger. Limiting velocity of high-dimensional random walk in random environment. Ann. Probab. 36 (2008) 728–738.
  • [2] F. Comets, N. Gantert and O. Zeitouni. Quenched, annealed and functional large deviations for one dimensional random walk in random environment. Probab. Theory Related Fields 118 (2000) 65–114.
  • [3] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. An invariance principle for reversible Markov processes with applications to random motions in random environments. J. Stat. Phys. 55 (1989) 787–855.
  • [4] A. Dembo and O. Zeitouni. Large Deviation Techniques and Applications, 2nd edition. Springer, New York, 1998.
  • [5] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 (1983) 183–212.
  • [6] A. Greven and F. den Hollander. Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381–1428.
  • [7] C. Kipnis and S. R. S. Varadhan. A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Comm. Math. Phys. 104 (1986) 1–19.
  • [8] S. M. Kozlov. The averaging method and walks in inhomogeneous environments. Russian Math. Surveys (Uspekhi Mat. Nauk) 40 (1985) 73–145.
  • [9] S. G. Krantz and H. R. Parks. The Implicit Function Theorem: History, Theory, and Applications. Birkhäuser, Boston, 2002.
  • [10] S. Olla. Homogenization of Diffusion Processes in Random Fields. Ecole Polytecnique, Palaiseau, 1994.
  • [11] G. Papanicolaou and S. R. S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. In Random Fields. J. Fritz and D. Szasz (eds). Janyos Bolyai Series. North-Holland, Amsterdam, 1981.
  • [12] J. Peterson. Limiting distributions and large deviations for random walks in random environments. Ph.D. thesis, Univ. Minnesota, 2008.
  • [13] J. Peterson and O. Zeitouni. On the annealed large deviation rate function for a multi-dimensional random walk in random environment. ALEA. To appear. Preprint, 2008. Available at arXiv:0812.3619.
  • [14] F. Rassoul-Agha. Large deviations for random walks in a mixing random environment and other (non-Markov) random walks. Comm. Pure Appl. Math. 57 (2004) 1178–1196.
  • [15] J. Rosenbluth. Quenched large deviations for multidimensional random walk in random environment: A variational formula. Ph.D. thesis, Courant Institute, New York Univ., 2006. Available at arXiv:0804.1444.
  • [16] A. S. Sznitman and M. Zerner. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999) 1851–1869.
  • [17] A. S. Sznitman. Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 2 (2000) 93–143.
  • [18] A. S. Sznitman. On a class of transient random walks in random environment. Ann. Probab. 29 (2001) 724–765.
  • [19] A. S. Sznitman. Lectures on random motions in random media. In Ten Lectures on Random Media. DMV-Lectures 32. Birkhäuser, Basel, 2002.
  • [20] S. R. S. Varadhan. Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56 (2003) 1222–1245.
  • [21] A. Yilmaz. Large deviations for random walk in a random environment. Ph.D. thesis, Courant Institute, New York Univ., 2008. Available at arXiv:0809.1227.
  • [22] A. Yilmaz. Quenched large deviations for random walk in a random environment. Comm. Pure Appl. Math. 62 (2009) 1033–1075.
  • [23] A. Yilmaz. On the equality of the quenched and averaged large deviation rate functions for high-dimensional ballistic random walk in a random environment. Preprint, 2009. Available at arXiv:0903.0410.
  • [24] O. Zeitouni. Random walks in random environments. J. Phys. A: Math. Gen. 39 (2006) R433–R464.
  • [25] M. P. W. Zerner. Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 (1998) 1446–1476.