The Annals of Probability

Lyapunov functions for random walks and strings in random environment

Francis Comets, Mikhail Menshikov, and Serguei Popov

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Abstract

We study two typical examples of countable Markov chains in random environment using the Lyapunov functions method: random walk and random string in random environment. In each case we construct an explicit Lyapunov function. Investigating the behavior of this function, we get the classification for recurrence, transience, ergodicity. We obtain new results for random strings in random environment, though we simply review well-known results for random walks using our approach.

Article information

Source
Ann. Probab. Volume 26, Number 4 (1998), 1433-1445.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1022855869

Mathematical Reviews number (MathSciNet)
MR1675023

Digital Object Identifier
doi:10.1214/aop/1022855869

Zentralblatt MATH identifier
0938.60065

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Random walk random string Lyapunov function random medium disordered systems product of random matrices Lyapunov exponents

Citation

Comets, Francis; Menshikov, Mikhail; Popov, Serguei. Lyapunov functions for random walks and strings in random environment. The Annals of Probability 26 (1998), no. 4, 1433--1445. doi:10.1214/aop/1022855869. http://projecteuclid.org/euclid.aop/1022855869.


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References

  • [1] Aspandiiarov, S., Iasnogorodski, R. and Menshikov, M. V. (1996). Passage-time moments for non-negative stochastic processes and an application to reflected random walk in a quadrant. Ann. Probab. 24 932-960.
  • [2] Bougerol, P. (1986). Oscillation de produits de matrices al´eatoires dont l'exposant de Lyapunov est nul. Lyapunov Exponents. Lecture Notes in Math. 1186 27-36. Springer, Berlin.
  • [3] Bougerol, P. and Lacroix, J. (1985). Products of Random Matrices with Application to Schr¨odinger Operators. Birkh¨auser, Boston.
  • [4] Comets, F., Menshikov, M. and Popov, S. (1997). One-dimensional branching random walk in a random environment: a classification. Markov Processes Related Fields. To appear.
  • [5] Cs´aki, E. (1978). On the lower limits of maxima and minima of Wiener process and partial sums.Wahrsch. Verw. Gebiete 43 205-221.
  • [6] Deheuvels, P. and Rev´esz, P. (1986). Simple random walk on the line in random environment. Probab. Theory Related Fields 72 215-230.
  • [7] Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge Univ. Press.
  • [8] Gajrat, A. S., Malyshev, V. A., Menshikov, M. V. and Pelih, K. D. (1995). Classification of Markov chains describing the evolution of a string of characters. Uspehi Mat. Nauk 50 5-24 (in Russian).
  • [9] Gajrat, A. S., Malyshev, V. A. and Zamyatin, A. A. (1995). Two-sided evolution of a random chain. Markov Processes Related Fields 1 281-316.
  • [10] Hu, Y. and Shi,(1998). The limits of Sinai's simple random walk in random environment. Ann. Probab. 26 1477-1521.
  • [11] Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Comp. Math. 30 145-168.
  • [12] Key, E. S. (1984). Recurrence and transience criteria for random walk in a random environment. Ann. Probab. 12 529-560.
  • [13] Kozlov, M. V. (1973). Random walk in one dimensional random medium. Theory Probab. Appl. 18 387-388.
  • [14] Ledrappier, F. (1984). Quelques propri´et´es des exposants caract´eristiques. Lecture Notes in Math. 1097 305-396. Springer, Berlin.
  • [15] Malyshev, V. A. (1997). Interacting strings. Uspehi Mat. Nauk 52 59-86 (in Russian).
  • [16] Meyn, S. P. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • [17] Oseledec, I. V. (1968). A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 197-231.
  • [18] R´ev´esz, P. (1990). Random Walk in Random and Nonrandom Environments. World Scientific, Teaneck, NJ.
  • [19] Sina¨i, Ya. G. (1982). The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27 256-268.
  • [20] Solomon, F. (1975). Random walks in a random environment. Ann. Probab. 3 1-31.