-
Previous Article
Serrin's regularity results for the incompressible liquid crystals system
- DCDS Home
- This Issue
-
Next Article
Infinite type flat surface models of ergodic systems
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 |
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. |
show all references
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. |
[1] |
Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete and Continuous Dynamical Systems, 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 and Heterogeneous Media, 2008, 3 (2) : 185-200. doi: 10.3934/nhm.2008.3.185 |
[3] |
Carey Caginalp. A survey of results on conservation laws with deterministic and random initial data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2043-2069. doi: 10.3934/dcdsb.2018225 |
[4] |
Guang-hui Cai. Strong laws for weighted sums of i.i.d. random variables. Electronic Research Announcements, 2006, 12: 29-36. |
[5] |
Dong Han Kim, Bing Li. Zero-one law of Hausdorff dimensions of the recurrent sets. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5477-5492. doi: 10.3934/dcds.2016041 |
[6] |
Hongyun Peng, Lizhi Ruan, Changjiang Zhu. Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis. Kinetic and Related Models, 2012, 5 (3) : 563-581. doi: 10.3934/krm.2012.5.563 |
[7] |
Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261 |
[8] |
Klaus Reiner Schenk-Hoppé. Random attractors--general properties, existence and applications to stochastic bifurcation theory. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 2006, 15 (1) : 73-86. doi: 10.3934/dcds.2006.15.73 |
[13] |
Joshua A. McGinnis, J. Douglas Wright. Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021100 |
[14] |
Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014 |
[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 and Continuous Dynamical Systems - S, 2017, 10 (2) : 289-311. doi: 10.3934/dcdss.2017014 |
[16] |
Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17 (1) : 101-128. doi: 10.3934/nhm.2021025 |
[17] |
Fabien Durand, Alejandro Maass. A note on limit laws for minimal Cantor systems with infinite periodic spectrum. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 745-750. doi: 10.3934/dcds.2003.9.745 |
[18] |
Juan C. Cortés, Sandra E. Delgadillo-Alemán, Roberto A. Kú-Carrillo, Rafael J. Villanueva. Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022079 |
[19] |
Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1 |
[20] |
Xiangdong Du, Martin Ostoja-Starzewski. On the scaling from statistical to representative volume element in thermoelasticity of random materials. Networks and Heterogeneous Media, 2006, 1 (2) : 259-274. doi: 10.3934/nhm.2006.1.259 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]