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On some random walks on \(\mathbb{Z}\) in random medium. (English) Zbl 1021.60034
The following model of random walks in random environment is considered. Let \((\Omega, {\mathcal F}, \mu, T)\) be an invertable dynamical system, i.e., \((\Omega, {\mathcal F}, \mu)\) is a probability space and \(T\) is a map on \(\Omega\), invertable and measurable together with its inverse. It is ergodic with respect to \(\mu\) and measure preserving. Given integers \(L, R \geq 1\), the set \(\Lambda\) is defined as \(\Lambda = \{ -L, -L+1 , \dots , 0 , \dots , R\}\). This is the set of possible jumps of the random walk on \(\mathbb Z\). The probabilities of such jumps are given by a collection \((p_z)_{z \in \Lambda}\) of positive random variables on \((\Omega, {\mathcal F}, \mu)\). They are supposed to obey the conditions: \(\sum_{z \in \Lambda }p_z (\omega) = 1\) \(\mu\)-almost surely; \(\exists \varepsilon >0\) such that \(\forall z \in \Lambda \setminus \{0\}\), one has \((p_z /p_R) \geq \varepsilon\) \(\mu\)-almost surely. The random walk \(\xi_n (\omega)\) is defined as a Markov chain by the transition probabilities \[ {\mathcal P}_\omega (\xi_{n+1} (\omega) = x + z \mid \xi_n (\omega) = x) = p_z (T^x \omega), \quad \;x \in {\mathbb Z}, \quad z \in \Lambda, \] and \(\xi_0 (\omega) = 0\). For this model with \(R=1\) the following results are obtained. A recurrence criterion expressed in terms of the sign of the maximal Lyapunov exponent of a certain random matrix is provided. An algorithm of calculating of this exponent is given. The existence of the absolutely continuous invariant measure for the Markov chain of “environments viewed from the particle” and of the nonzero drift (in the transient case) are characterized. A sufficient condition for the validity of the central limit theorem in the transient case is given.

MSC:
60G50 Sums of independent random variables; random walks
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
60K37 Processes in random environments
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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[1] ALILI, S. (1999). Asy mptotic behaviour for random walks in random environments. J. Appl. Probab. 36 334-349. · Zbl 0946.60046 · doi:10.1239/jap/1032374457
[2] ARNOLD, V. I. (1961). Small denominators I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25 21-86. · Zbl 0152.41905
[3] ATKINSON, G. (1976). Recurrence of co-cy cles and random walks. J. London Math. Soc. (2) 13 486-488. · Zbl 0342.60049 · doi:10.1112/jlms/s2-13.3.486
[4] BERNASCONI, J. and SCHNEIDER, T., eds. (1981). physics in One Dimension. Springer, Berlin.
[5] BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[6] BREMONT, J. (2001). Marches aléatoires sur Z en milieu gibbsien et théorème de Sinaï.
[7] BREMONT, J. (2001). On the recurrence of random walks on Z in random medium. C. R. Acad. Sci. Paris Sér. I Math. 333 1-6. · Zbl 0995.60062 · doi:10.1016/S0764-4442(01)02173-5
[8] BROWN, B. M. (1971). Martingale central limit theorems. Ann. Math. Statist. 42 59-66. · Zbl 0218.60048 · doi:10.1214/aoms/1177693494
[9] BULy CHEVA, O. G. and MOLCHANOV, S. A. (1986). Averaged description of one-dimensional random media. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1986 33-38, 119. · Zbl 0599.60065
[10] CONZE, J.-P. (1976). Remarques sur les transformations cy lindriques et les équations fonctionnelles. Séminaire de Probabilités 1 13. Springer, Berlin.
[11] CONZE, J.-P. and GUIVARC’H, Y. (2000). Marches en milieu aléatoire et mesures quasiinvariantes pour un sy stème dy namique. Colloq. Math. 84/85 457-480.
[12] DERRIENNIC, Y. (1999). Sur la récurrence des marches aléatoires unidimensionnelles en environnement aléatoire. C. R. Acad. Sci. Paris Sér. I Math. 329 65-70. · Zbl 0938.60060 · doi:10.1016/S0764-4442(99)80463-7
[13] GANTERT, N. and ZEITOUNI, O. (1999). Large deviations for one-dimensional random walk in a random environment-a survey. Boly ai Soc. Math. Stud. 127-165. · Zbl 0953.60009
[14] GREVEN, A. and HOLLANDER, F. (1994). Large deviations for a random walk in random environment. Ann. Probab. 22 1381-1428. · Zbl 0820.60054 · doi:10.1214/aop/1176988607
[15] HENNION, H. (1997). Limit theorems for products of positive random matrices. Ann. Probab. 25 1545-1587. · Zbl 0903.60027 · doi:10.1214/aop/1023481103
[16] HU, Y. and SHI, Z. (1998). The limits of Sinai’s simple random walk in random environment. Ann. Probab. 26 1477-1521. · Zbl 0936.60088 · doi:10.1214/aop/1022855871
[17] KESTEN, H., KOZLOV, M. V. and SPITZER, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30 145-168. · Zbl 0388.60069 · numdam:CM_1975__30_2_145_0 · eudml:89251
[18] KEY, E. S. (1984). Recurrence and transience criteria for random walk in a random environment. Ann. Probab. 12 529-560. · Zbl 0545.60066 · doi:10.1214/aop/1176993304
[19] KOZLOV, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61-120, 238. · Zbl 0615.60063 · doi:10.1070/RM1985v040n02ABEH003558
[20] KOZLOV, S. M. and MOLCHANOV, S. A. (1984). Conditions for the applicability of the central limit theorem to random walks on a lattice. Dokl. Akad. Nauk SSSR 278 531-534. · Zbl 0603.60020
[21] LËTCHIKOV, A. V. (1989). A limit theorem for a recurrent random walk in a random environment. Dokl. Akad. Nauk SSSR 304 25-28. · Zbl 0678.60060
[22] LËTCHIKOV, A. V. (1989). Localization of One-Dimensional Random Walks in Random Environments. Routledge, London. · Zbl 0684.60056
[23] LËTCHIKOV, A. V. (1992). A criterion for the applicability of the central limit theorem to onedimensional random walks in random environments. Teor. Veroy atnost. i Primenen. 37 576-580. · Zbl 0787.60088 · doi:10.1137/1137107
[24] MOLCHANOV, S. (1994). Lectures on random media. Lectures on Probability Theory. Lecture Notes in Math. 1581 242-411. Springer, Berlin. · Zbl 0814.60093
[25] OSELEDEC, V. I. (1968). A multiplicative ergodic theorem: characteristic Liapounov, exponents of dy namical sy stems. Trudy Moskov. Mat. Obshch. 19 179-210. · Zbl 0236.93034
[26] RAUGI, A. (1997). Théorème ergodique multiplicatif. Produits de matrices aléatoires indépendantes. Fascicule de Probabilités 43. · Zbl 0947.60008
[27] SINA I, Y. G. (1982). The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroy atnost. i Primenen. 27 247-258. · Zbl 0497.60065
[28] SOLOMON, F. (1975). Random walks in a random environment. Ann. Probab. 3 1-31. · Zbl 0305.60029 · doi:10.1214/aop/1176996444
[29] ZEITOUNI, O. (2001). St. Flour lecture notes on random walks in random environment. Technical report. Available at www-ee.technion.ac.il/ zeitouni.
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