# American Institute of Mathematical Sciences

October  2016, 36(10): 5555-5578. doi: 10.3934/dcds.2016044

## Deterministically driven random walks in a random environment on $\mathbb{Z}$

 1 The Department of Mathematics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, GU2 7XH, United Kingdom

Received  October 2015 Revised  February 2016 Published  July 2016

We introduce the concept of a deterministic walk in a deterministic environment on a state space $S$ (DWDE), focusing on the case where $S$ is countable. For the deterministic walk in a fixed environment we establish properties analogous to those found in Markov chain theory, but for systems that do not in general have the Markov property (in the stochastic process sense). In particular, we establish hypotheses ensuring that a DWDE on $\mathbb{Z}$ is either recurrent or transient. An immediate consequence of this result is that a symmetric DWDE on $\mathbb{Z}$ is recurrent. Moreover, in the transient case, we show that the probability that the DWDE diverges to $+ \infty$ is either 0 or 1. In certain cases we compute the direction of divergence in the transient case.
Citation: Colin Little. Deterministically driven random walks in a random environment on $\mathbb{Z}$. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5555-5578. doi: 10.3934/dcds.2016044
##### References:
 [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, American Mathematical Society, 1997. doi: 10.1090/surv/050.  Google Scholar [2] J. Aaronson, M. Denker and M. Urbanski, Ergodic theory for Markov fibred systems and parabolic rational maps, Transactions of the American Mathematical Society, 337 (1993), 495-548. doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar [3] J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stochastic Dynamics, 1 (2001), 193-237. doi: 10.1142/S0219493701000114.  Google Scholar [4] E. Bolthausen and I. Goldsheid, Recurrence and transience of random walks in random environments on a strip, Communications in Mathematical Physics, 214 (2000), 429-447. doi: 10.1007/s002200000279.  Google Scholar [5] J. Bremont, Behaviour of random walks on $\mathbbZ$ in a Gibbsian medium, C. R. Acad. Sci. Serie, 338 (2004), 895-898. doi: 10.1016/j.crma.2004.03.030.  Google Scholar [6] L. A. Bunimovich, Y. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiard, Uspekhi Mat. Nauk, 46 (1991), 43-92. doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar [7] G. Cristadoro, M. Lenci and M. Seri, Recurrence for quenched random Lorentz tubes, Chaos, 20 (2010), 023115, 7 pp; erratum at Chaos, 20 (2010), 049903, 1 pp. doi: 10.1063/1.3405290.  Google Scholar [8] G. Cristadoro, M. Degli Esposti, M. Lenci and M. Seri, Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two, Journal of Statistical Physics, 144 (2011), 124-138. doi: 10.1007/s10955-011-0244-5.  Google Scholar [9] D. Dolgopyat and L. Koralov, Motion in a random force field, Nonlinearity, 22 (2009), 187-211. doi: 10.1088/0951-7715/22/1/010.  Google Scholar [10] D. Dolgopyat, D. Szász and T. Varjú, Limit Theorems for Locally Perturbed Lorentz processes, Duke Math. J., 148 (2009), 459-499. doi: 10.1215/00127094-2009-031.  Google Scholar [11] H. Kesten, M. V. Koslov and F. Spitzer, A limit law for random walk in a random environment, Compositio Mathematica, 30 (1975), 145-168.  Google Scholar [12] E. Key, Recurrence and transience criteria for random walk in a random environment, Annals of Probability, 12 (1984), 529-560. doi: 10.1214/aop/1176993304.  Google Scholar [13] M. V. Koslov, A random walk on a line with stochastic structure, Theory of Probability and its Applications, 18 (1973), 406-408. Google Scholar [14] M. Lenci and S. Troubetzkoy, Infinite-horizon Lorentz tubes and gases: Recurrence and ergodic properties, Physica D: Nonlinear Phenomena, 240 (2011), 1510-1515. doi: 10.1016/j.physd.2011.06.020.  Google Scholar [15] C. Little, Deterministically Driven Random Walks in Random Environment, Ph.D thesis, University of Surrey, 2013. Google Scholar [16] C. Little, Deterministically driven random walks on a finite state space, Dynamical Systems: An International Journal, 30 (2015), 200-207. doi: 10.1080/14689367.2014.993926.  Google Scholar [17] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Communications in Mathematical Physics, 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5.  Google Scholar [18] I. Melbourne and M. Nicol, A vector-valued almost sure invariance principle for hyperbolic dynamical systems, Annals of Probability, 37 (2009), 478-505. doi: 10.1214/08-AOP410.  Google Scholar [19] T. Simula and M. Stenlund, Deterministic walks in quenched random environments of chaotic map, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 245101 (14 pp). doi: 10.1088/1751-8113/42/24/245101.  Google Scholar [20] Ya. G. Sinai, Dynamical systems with elastic reflections: Ergodic properties of dispersing billiards, Russian Mathematical Surveys, 25 (1970), 141-192.  Google Scholar [21] Ya. G. Sinai, Limit behaviour of one-dimensional random walks in random environments, Theory of Probability and its Applications, 27 (1982), 247-258. Google Scholar [22] F. Solomon, Random walks in a random environment, Annals of Probability, 3 (1975), 1-31. doi: 10.1214/aop/1176996444.  Google Scholar [23] M. Stenlund, A vector-valued almost sure invariance principle for Sinai billiards with random scatterers, Comm. Math. Phys., 325 (2014), 879-916, http://arxiv.org/abs/1210.0902. doi: 10.1007/s00220-013-1870-3.  Google Scholar [24] A. S. Sznitman, Topics in random walks in random environment, in School and Conference on Probability Theory, ICTP Lecture Notes Series, Trieste, 17 (2004), 203-266.  Google Scholar [25] O. Zeitouni, Random walks in random environment, in XXXI Summer School in Probability, St. Flour (2001). Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York. 1837 (2004) 189-312. doi: 10.1007/978-3-540-39874-5_2.  Google Scholar

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##### References:
 [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, American Mathematical Society, 1997. doi: 10.1090/surv/050.  Google Scholar [2] J. Aaronson, M. Denker and M. Urbanski, Ergodic theory for Markov fibred systems and parabolic rational maps, Transactions of the American Mathematical Society, 337 (1993), 495-548. doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar [3] J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stochastic Dynamics, 1 (2001), 193-237. doi: 10.1142/S0219493701000114.  Google Scholar [4] E. Bolthausen and I. Goldsheid, Recurrence and transience of random walks in random environments on a strip, Communications in Mathematical Physics, 214 (2000), 429-447. doi: 10.1007/s002200000279.  Google Scholar [5] J. Bremont, Behaviour of random walks on $\mathbbZ$ in a Gibbsian medium, C. R. Acad. Sci. Serie, 338 (2004), 895-898. doi: 10.1016/j.crma.2004.03.030.  Google Scholar [6] L. A. Bunimovich, Y. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiard, Uspekhi Mat. Nauk, 46 (1991), 43-92. doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar [7] G. Cristadoro, M. Lenci and M. Seri, Recurrence for quenched random Lorentz tubes, Chaos, 20 (2010), 023115, 7 pp; erratum at Chaos, 20 (2010), 049903, 1 pp. doi: 10.1063/1.3405290.  Google Scholar [8] G. Cristadoro, M. Degli Esposti, M. Lenci and M. Seri, Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two, Journal of Statistical Physics, 144 (2011), 124-138. doi: 10.1007/s10955-011-0244-5.  Google Scholar [9] D. Dolgopyat and L. Koralov, Motion in a random force field, Nonlinearity, 22 (2009), 187-211. doi: 10.1088/0951-7715/22/1/010.  Google Scholar [10] D. Dolgopyat, D. Szász and T. Varjú, Limit Theorems for Locally Perturbed Lorentz processes, Duke Math. J., 148 (2009), 459-499. doi: 10.1215/00127094-2009-031.  Google Scholar [11] H. Kesten, M. V. Koslov and F. Spitzer, A limit law for random walk in a random environment, Compositio Mathematica, 30 (1975), 145-168.  Google Scholar [12] E. Key, Recurrence and transience criteria for random walk in a random environment, Annals of Probability, 12 (1984), 529-560. doi: 10.1214/aop/1176993304.  Google Scholar [13] M. V. Koslov, A random walk on a line with stochastic structure, Theory of Probability and its Applications, 18 (1973), 406-408. Google Scholar [14] M. Lenci and S. Troubetzkoy, Infinite-horizon Lorentz tubes and gases: Recurrence and ergodic properties, Physica D: Nonlinear Phenomena, 240 (2011), 1510-1515. doi: 10.1016/j.physd.2011.06.020.  Google Scholar [15] C. Little, Deterministically Driven Random Walks in Random Environment, Ph.D thesis, University of Surrey, 2013. Google Scholar [16] C. Little, Deterministically driven random walks on a finite state space, Dynamical Systems: An International Journal, 30 (2015), 200-207. doi: 10.1080/14689367.2014.993926.  Google Scholar [17] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Communications in Mathematical Physics, 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5.  Google Scholar [18] I. Melbourne and M. Nicol, A vector-valued almost sure invariance principle for hyperbolic dynamical systems, Annals of Probability, 37 (2009), 478-505. doi: 10.1214/08-AOP410.  Google Scholar [19] T. Simula and M. Stenlund, Deterministic walks in quenched random environments of chaotic map, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 245101 (14 pp). doi: 10.1088/1751-8113/42/24/245101.  Google Scholar [20] Ya. G. Sinai, Dynamical systems with elastic reflections: Ergodic properties of dispersing billiards, Russian Mathematical Surveys, 25 (1970), 141-192.  Google Scholar [21] Ya. G. Sinai, Limit behaviour of one-dimensional random walks in random environments, Theory of Probability and its Applications, 27 (1982), 247-258. Google Scholar [22] F. Solomon, Random walks in a random environment, Annals of Probability, 3 (1975), 1-31. doi: 10.1214/aop/1176996444.  Google Scholar [23] M. Stenlund, A vector-valued almost sure invariance principle for Sinai billiards with random scatterers, Comm. Math. Phys., 325 (2014), 879-916, http://arxiv.org/abs/1210.0902. doi: 10.1007/s00220-013-1870-3.  Google Scholar [24] A. S. Sznitman, Topics in random walks in random environment, in School and Conference on Probability Theory, ICTP Lecture Notes Series, Trieste, 17 (2004), 203-266.  Google Scholar [25] O. Zeitouni, Random walks in random environment, in XXXI Summer School in Probability, St. Flour (2001). Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York. 1837 (2004) 189-312. doi: 10.1007/978-3-540-39874-5_2.  Google Scholar
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