American Institute of Mathematical Sciences

Advanced
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

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 & 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]

Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689
[2]

Jan Lorenz, Stefano Battiston. Systemic risk in a network fragility model analyzed with probability density evolution of persistent random walks. Networks & Heterogeneous Media, 2008, 3 (2) : 185-200. doi: 10.3934/nhm.2008.3.185
[3]

Guang-hui Cai. Strong laws for weighted sums of i.i.d. random variables. Electronic Research Announcements, 2006, 12: 29-36.

[4]

Carey Caginalp. A survey of results on conservation laws with deterministic and random initial data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2043-2069. doi: 10.3934/dcdsb.2018225
[5]

Dong Han Kim, Bing Li. Zero-one law of Hausdorff dimensions of the recurrent sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5477-5492. doi: 10.3934/dcds.2016041
[6]

Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261
[7]

Hongyun Peng, Lizhi Ruan, Changjiang Zhu. Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis. Kinetic & Related Models, 2012, 5 (3) : 563-581. doi: 10.3934/krm.2012.5.563
[8]

Klaus Reiner Schenk-Hoppé. Random attractors--general properties, existence and applications to stochastic bifurcation theory. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 99-130. doi: 10.3934/dcds.1998.4.99
[9]

Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics & Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020
[10]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[11]

Christina Knox, Amir Moradifam. Electrical networks with prescribed current and applications to random walks on graphs. Inverse Problems & Imaging, 2019, 13 (2) : 353-375. doi: 10.3934/ipi.2019018
[12]

V. Chaumoître, M. Kupsa. k-limit laws of return and hitting times. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73
[13]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[14]

Xiangdong Du, Martin Ostoja-Starzewski. On the scaling from statistical to representative volume element in thermoelasticity of random materials. Networks & Heterogeneous Media, 2006, 1 (2) : 259-274. doi: 10.3934/nhm.2006.1.259
[15]

Kumiko Hattori, Noriaki Ogo, Takafumi Otsuka. A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 289-311. doi: 10.3934/dcdss.2017014
[16]

Fabien Durand, Alejandro Maass. A note on limit laws for minimal Cantor systems with infinite periodic spectrum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 745-750. doi: 10.3934/dcds.2003.9.745
[17]

James Nolen, Jack Xin. KPP fronts in a one-dimensional random drift. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 421-442. doi: 10.3934/dcdsb.2009.11.421
[18]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[19]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[20]

Martin Oberlack, Andreas Rosteck. New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 451-471. doi: 10.3934/dcdss.2010.3.451

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences