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Random Walks in Random Environment

  • Tomasz Komorowski
  • Claudio Landim
  • Stefano Olla
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 345)

Abstract

Random walks in random environment constitute the best example to apply the theory developed in Chap.  2. In order to keep notation and results consistent with the previous chapter, we discuss only the time continuous case. By exploiting the translation invariance of the random environment, the position of the random walk is seen as an additive functional of the environment as seen from the particle process. Double-stochastic (or bi-stochastic) random walks are the basic examples where the ergodic stationary measure for the environment process is known explicitly. Random walks with random conductances (so called bond diffusions) are the simplest example and the resulting environment process is reversible. If the condition of a finite cycle decomposition is satisfied the sector condition for the environment process holds. For more general bi-stochastic environments we can prove the central limit theorem in dimension 3 or higher if a mixing condition is satisfied. In dimension 1 the sector condition is always satisfied if the drift has null average.

Keywords

Random Walk Central Limit Theorem Sector Condition Dirichlet Form Random Environment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Anshelevich VV, Khanin KM, Sinaĭ YG (1982) Symmetric random walks in random environments. Commun Math Phys 85(3):449–470 zbMATHCrossRefGoogle Scholar
  2. Barlow MT (2004) Random walks on supercritical percolation clusters. Ann Probab 32(4):3024–3084 MathSciNetzbMATHCrossRefGoogle Scholar
  3. Barlow MT, Deuschel JD (2010) Invariance principle for the random conductance model with unbounded conductances. Ann Probab 38(1):234–276 MathSciNetzbMATHCrossRefGoogle Scholar
  4. Berger N, Biskup M (2007) Quenched invariance principle for simple random walk on percolation clusters. Probab Theory Relat Fields 137(1–2):83–120 MathSciNetzbMATHGoogle Scholar
  5. Bezuidenhout C, Grimmett G (1999) A central limit theorem for random walks in random labyrinths. Ann Inst Henri Poincaré Probab Stat 35(5):631–683 MathSciNetzbMATHCrossRefGoogle Scholar
  6. Biskup M, Prescott TM (2007) Functional CLT for random walk among bounded random conductances. Electron J Probab 12(49):1323–1348 (electronic) MathSciNetzbMATHGoogle Scholar
  7. Dacunha-Castelle D, Duflo M (1986) Probability and statistics, vol II. Springer, New York. Translated from the French by David McHale CrossRefGoogle Scholar
  8. De Masi A, Ferrari PA, Goldstein S, Wick WD (1989) An invariance principle for reversible Markov processes. Applications to random motions in random environments. J Stat Phys 55(3–4):787–855 zbMATHCrossRefGoogle Scholar
  9. Deuschel JD, Kösters H (2008) The quenched invariance principle for random walks in random environments admitting a bounded cycle representation. Ann Inst Henri Poincaré Probab Stat 44(3):574–591 zbMATHCrossRefGoogle Scholar
  10. Dolgopyat D, Keller G, Liverani C (2008) Random walk in Markovian environment. Ann Probab 36(5):1676–1710 MathSciNetzbMATHCrossRefGoogle Scholar
  11. Doukhan P (1994) Mixing: properties and examples. Lecture notes in statistics, vol 85. Springer, New York zbMATHGoogle Scholar
  12. Goldstein S (1995) Antisymmetric functionals of reversible Markov processes. Ann Inst Henri Poincaré Probab Stat 31(1):177–190 zbMATHGoogle Scholar
  13. Ibragimov IA, Linnik YV (1971) Independent and stationary sequences of random variables: with a supplementary chapter by IA Ibragimov and VV Petrov. Wolters-Noordhoff Publishing, Groningen. Translation from the Russian edited by JFC Kingman Google Scholar
  14. Komorowski T, Olla S (2003a) A note on the central limit theorem for two-fold stochastic random walks in a random environment. Bull Pol Acad Sci, Math 51(2):217–232 MathSciNetzbMATHGoogle Scholar
  15. Künnemann R (1983) The diffusion limit for reversible jump processes on Z d with ergodic random bond conductivities. Commun Math Phys 90(1):27–68 zbMATHCrossRefGoogle Scholar
  16. Leoni G (2009) A first course in Sobolev spaces. Graduate studies in mathematics, vol 105. Am Math Soc, Providence zbMATHGoogle Scholar
  17. Mathieu P (2006) Carne-Varopoulos bounds for centered random walks. Ann Probab 34(3):987–1011 MathSciNetzbMATHCrossRefGoogle Scholar
  18. Mathieu P (2008) Quenched invariance principles for random walks with random conductances. J Stat Phys 130(5):1025–1046 MathSciNetzbMATHCrossRefGoogle Scholar
  19. Mathieu P, Piatnitski A (2007) Quenched invariance principles for random walks on percolation clusters. Proc R Soc Lond, Ser A, Math Phys Eng Sci 463(2085):2287–2307 MathSciNetzbMATHCrossRefGoogle Scholar
  20. Oelschläger K (1988) Homogenization of a diffusion process in a divergence-free random field. Ann Probab 16(3):1084–1126 MathSciNetzbMATHCrossRefGoogle Scholar
  21. Osada H, Saitoh T (1995) An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains. Probab Theory Relat Fields 101(1):45–63 MathSciNetzbMATHCrossRefGoogle Scholar
  22. Rassoul-Agha F, Seppäläinen T (2005) An almost sure invariance principle for random walks in a space-time random environment. Probab Theory Relat Fields 133(3):299–314 zbMATHCrossRefGoogle Scholar
  23. Sidoravicius V, Sznitman AS (2004) Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab Theory Relat Fields 129(2):219–244 MathSciNetzbMATHCrossRefGoogle Scholar
  24. Sznitman AS (2004) Topics in random walks in random environment. In: School and conference on probability theory. ICTP lect notes, vol XVII. Abdus Salam Int Cent Theoret Phys, Trieste, pp 203–266 (electronic) Google Scholar
  25. Tanemura H (1993) Central limit theorem for a random walk with random obstacles in ℝd. Ann Probab 21(2):936–960 MathSciNetzbMATHCrossRefGoogle Scholar
  26. Yosida K (1995) Functional analysis. Classics in mathematics. Springer, Berlin. Reprint of the sixth 1980 edition Google Scholar
  27. Zeitouni O (2004) Random walks in random environment. In: Lectures on probability theory and statistics. Lecture notes in math, vol 1837. Springer, Berlin, pp 189–312 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tomasz Komorowski
    • 1
    • 2
  • Claudio Landim
    • 3
    • 4
  • Stefano Olla
    • 5
  1. 1.Institute of MathematicsMaria Curie-Skłodowska UniversityLublinPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  3. 3.Instituto de Matemática (IMPA)Rio de JaneiroBrazil
  4. 4.CNRS UMR 6085Université de RouenSaint-Étienne-du-RouvrayFrance
  5. 5.CEREMADE, CNRS UMR 7534Université Paris-DauphineParisFrance

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