The Annals of Probability

Random walk in Markovian environment

Dmitry Dolgopyat, Gerhard Keller, and Carlangelo Liverani

Full-text: Open access

Abstract

We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on ℤd. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.

Article information

Source
Ann. Probab., Volume 36, Number 5 (2008), 1676-1710.

Dates
First available in Project Euclid: 11 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1221138763

Digital Object Identifier
doi:10.1214/07-AOP369

Mathematical Reviews number (MathSciNet)
MR2440920

Zentralblatt MATH identifier
1192.60110

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems 37H99: None of the above, but in this section 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Central limit theorem random walk random environment Markov process

Citation

Dolgopyat, Dmitry; Keller, Gerhard; Liverani, Carlangelo. Random walk in Markovian environment. Ann. Probab. 36 (2008), no. 5, 1676--1710. doi:10.1214/07-AOP369. https://projecteuclid.org/euclid.aop/1221138763


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References

  • [1] Bandyopadhyay, A. and Zeitouni, O. (2006). Random walk in dynamic Markovian random environment. ALEA Lat. Amer. J. Probab. Statist. 1 205–224.
  • [2] Boldrighini, C., Minlos, R. A. and Pellegrinotti, A. (1994). Central limit theorem for the random walk of one or two particles in a random environment. Adv. Soviet Math. 20 21–75.
  • [3] Boldrighini, C., Minlos, R. A. and Pellegrinotti, A. (2000). Random walk in a fluctuating random environment with Markov evolution. In On Dobrushin’s Way. From Probability Theory to Statistical Physics (R. A. Minlos, S. Shlosman and Yu. M. Suhov, eds.) Amer. Math. Soc., 13–35. Providence, RI.
  • [4] Boldrighini, C., Minlos, R. A. and Pellegrinotti, A. (2004). Random walks in quenched i.i.d. space–time random environment are always a.s. diffusive. Probab. Theory Related Fields 129 133–156.
  • [5] Bolthausen, E. and Sznitman, A.-S. (2002). On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 345–375.
  • [6] Conlon, J. G. and Song, R. (1999). Gaussian limit theorems for diffusion processes and an application. Stochastic Process. Appl. 81 103–128.
  • [7] Comets, F. and Zeitouni, O. (2005). Gaussian fluctuations for random walks in random mixing environments. Israel J. Math. 148 87–113.
  • [8] Derriennic, Y. and Lin, M. (2001). Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123 93–130.
  • [9] Derriennic, Y. and Lin, M. (2003). The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 73–76.
  • [10] Fannjiang, A. and Komorowski, T. (2001). Invariance principle for a diffusion in a Markov field. Bull. Polish Acad. Sci. Math. 49 45–65.
  • [11] Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 739–741.
  • [12] Grimmett, G. and Stirzaker, D. (2001). Probability and Random Processes, 3rd ed. Oxford Univ. Press.
  • [13] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York.
  • [14] Komorowski, T. and Olla, S. (2001). On homogenization of time-dependent random flows. Probab. Theory Related Fields 121 98–116.
  • [15] Keller, G. and Liverani, C. (2006). Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Comm. Math. Phys. 262 33–50.
  • [16] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • [17] Landim, C., Olla, S. and Yau, H. T. (1998). Convection–diffusion equation with space–time ergodic random flow. Probab. Theory Related Fields 112 203–220.
  • [18] Liverani, C. (1996). Central limit theorem for deterministic systems. In International Conference on Dynamical Systems (Montevideo, 1995). A Tribute to Ricardo Mane. (F. Ledrappier, J. Levovicz and S. Newhouse, eds.) 56–75. Longman, Harlow.
  • [19] Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713–724.
  • [20] Rassoul-Agha, F. and Seppalainen, T. (2005). An almost sure invariance principle for random walks in a space–time i.i.d. random environment. Probab. Theory Related Fields 133 299–314.
  • [21] Rassoul-Agha, F. and Seppalainen, T. (2005). An almost sure invariance principle for additive functionals of Markov chains. Preprint. Available at http://arxiv.org/abs/math/0411603v2.
  • [22] Stannat, W. (2004). A remark on the CLT for random walks in a random environment. Probab. Theory Related Fields 130 377–387.
  • [23] Sinai, Y. G. (1993). A random walk with a random potential. Teor. Veroyatnost. i Primenen. 38 457–460. [Translation in Theory Probab. Appl. 38 (1993) 382–385.]
  • [24] Sinai, Y. G. (1995). A remark concerning random walks with random potentials. Fund. Math. 147 173–180.
  • [25] Sznitman, A.-S. (2004). Topics in random walk in random environment. In School and Conference on Probability Theory 203–266. Electronic.
  • [26] Zeitouni, O. (2004). Random walks in random environment. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin.