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

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

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

Colin Little ^1,

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

Abstract
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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.

Keywords: zero-one laws, statistical limit laws, Relations with probability theory and stochastic processes, random walks in a random environment..

Mathematics Subject Classification: Primary: 37A50; Secondary: 60F2.

Citation: Colin Little. Deterministically driven random walks in a random environment on $\mathbb{Z}$. Discrete and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	M. V. Koslov, A random walk on a line with stochastic structure, Theory of Probability and its Applications, 18 (1973), 406-408.
[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.
[15]	C. Little, Deterministically Driven Random Walks in Random Environment, Ph.D thesis, University of Surrey, 2013.
[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.
[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.
[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.
[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.
[20]	Ya. G. Sinai, Dynamical systems with elastic reflections: Ergodic properties of dispersing billiards, Russian Mathematical Surveys, 25 (1970), 141-192.
[21]	Ya. G. Sinai, Limit behaviour of one-dimensional random walks in random environments, Theory of Probability and its Applications, 27 (1982), 247-258.
[22]	F. Solomon, Random walks in a random environment, Annals of Probability, 3 (1975), 1-31. doi: 10.1214/aop/1176996444.
[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.
[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.
[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.

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References:

[1]	J. Aaronson, An Introduction to Infinite Ergodic Theory, American Mathematical Society, 1997. doi: 10.1090/surv/050.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	M. V. Koslov, A random walk on a line with stochastic structure, Theory of Probability and its Applications, 18 (1973), 406-408.
[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.
[15]	C. Little, Deterministically Driven Random Walks in Random Environment, Ph.D thesis, University of Surrey, 2013.
[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.
[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.
[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.
[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.
[20]	Ya. G. Sinai, Dynamical systems with elastic reflections: Ergodic properties of dispersing billiards, Russian Mathematical Surveys, 25 (1970), 141-192.
[21]	Ya. G. Sinai, Limit behaviour of one-dimensional random walks in random environments, Theory of Probability and its Applications, 27 (1982), 247-258.
[22]	F. Solomon, Random walks in a random environment, Annals of Probability, 3 (1975), 1-31. doi: 10.1214/aop/1176996444.
[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.
[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.
[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.

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