# 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 - A, 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.  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.  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.  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.  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.  doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar [7] G. Cristadoro, M. Lenci and M. Seri, Recurrence for quenched random Lorentz tubes,, Chaos, 20 (2010).  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.  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.  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.  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.   Google Scholar [12] E. Key, Recurrence and transience criteria for random walk in a random environment,, Annals of Probability, 12 (1984), 529.  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.   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.  doi: 10.1016/j.physd.2011.06.020.  Google Scholar [15] C. Little, Deterministically Driven Random Walks in Random Environment,, Ph.D thesis, (2013).   Google Scholar [16] C. Little, Deterministically driven random walks on a finite state space,, Dynamical Systems: An International Journal, 30 (2015), 200.  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.  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.  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).  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.   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.   Google Scholar [22] F. Solomon, Random walks in a random environment,, Annals of Probability, 3 (1975), 1.  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.  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, 17 (2004), 203.   Google Scholar [25] O. Zeitouni, Random walks in random environment,, in XXXI Summer School in Probability, 1837 (2001), 189.  doi: 10.1007/978-3-540-39874-5_2.  Google Scholar

show all references

##### 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.  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.  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.  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.  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.  doi: 10.1070/RM1991v046n04ABEH002827.  Google Scholar [7] G. Cristadoro, M. Lenci and M. Seri, Recurrence for quenched random Lorentz tubes,, Chaos, 20 (2010).  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.  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.  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.  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.   Google Scholar [12] E. Key, Recurrence and transience criteria for random walk in a random environment,, Annals of Probability, 12 (1984), 529.  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.   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.  doi: 10.1016/j.physd.2011.06.020.  Google Scholar [15] C. Little, Deterministically Driven Random Walks in Random Environment,, Ph.D thesis, (2013).   Google Scholar [16] C. Little, Deterministically driven random walks on a finite state space,, Dynamical Systems: An International Journal, 30 (2015), 200.  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.  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.  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).  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.   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.   Google Scholar [22] F. Solomon, Random walks in a random environment,, Annals of Probability, 3 (1975), 1.  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.  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, 17 (2004), 203.   Google Scholar [25] O. Zeitouni, Random walks in random environment,, in XXXI Summer School in Probability, 1837 (2001), 189.  doi: 10.1007/978-3-540-39874-5_2.  Google Scholar
 [1] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [2] Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 [3] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352 [4] Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329 [5] Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 [6] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168 [7] Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 [8] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [9] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [10] Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158 [11] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [12] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [13] Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051 [14] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [15] Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044 [16] Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344 [17] Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270 [18] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [19] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [20] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

2019 Impact Factor: 1.338