June  2021, 41(6): 2891-2905. doi: 10.3934/dcds.2020390

A symmetric Random Walk defined by the time-one map of a geodesic flow

Av. Presidente Antônio Carlos 6627, Belo Horizonte-MG, BR31270-901

* Corresponding author: Pablo D. Carrasco

Received  August 2020 Revised  October 2020 Published  June 2021 Early access  December 2020

In this note we consider a symmetric Random Walk defined by a $ (f, f^{-1}) $ Kalikow type system, where $ f $ is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the Random Walk are deduced in these cases.

Citation: Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390
References:
[1]

D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[2]

A. AvilaM. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows, Journal of the European Mathematical Society, 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.  Google Scholar

[3]

A. Candel and L. Conlon, Foliations I, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/gsm/023.  Google Scholar

[4]

J.-P. Conze and Y. Guivarc'h, Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique, Colloquium Mathematicum, 84 (2000), 457-480.  doi: 10.4064/cm-84/85-2-457-480.  Google Scholar

[5]

D. Dolgopyat, B. Fayad, and M. Saprykina, Erratic behavior for 1-dimensional Random Walks in a Liouville quasi-periodic environment., preprint, 2019, arXiv: 1901.10709. Google Scholar

[6]

M. Gorodin and B. Lifsic, Central limit theorem for stationary Markov processes, In Third Vilnius Conference on Probability and Statistics, volume 1, 1981,147–148. Google Scholar

[7] B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University Press, New York, 2003.  doi: 10.1017/CBO9780511998188.  Google Scholar
[8]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A Survey of Partially Hyperbolic Dynamics, In Partially Hyperbolic Dynamics, Laminations and Teichmüller Flow, Fields Institute Communications, vol. 51, 2007, 35–88.  Google Scholar

[9]

M. W. Hirsch, C. C. Pugh, and M. I. Shub, Invariant Manifolds, Springer Berlin Heidelberg, 1977. Google Scholar

[10]

V. Kaloshin and Y. Sinai, Simple random walks along orbits of Anosov diffeomorphisms, Tr. Mat. Inst. Steklova, 228 (2000), 236-245.   Google Scholar

[11]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210.  doi: 10.4310/MRL.1996.v3.n2.a6.  Google Scholar

[12]

Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

[13]

C. Kipnis and S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications Math. Physics, 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[14]

J. Neveu and A. Feinstein, Mathematical Foundations of the Calculus of Probability, Holden-Day, Inc. San Francisco, Calif.-London-Amsterdam, 1965.  Google Scholar

[15]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, Zürich, 2004. doi: 10.4171/003.  Google Scholar

[16]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS), 2 (2000), 1-52.  doi: 10.1007/s100970050013.  Google Scholar

[17]

F. Rodriguez-HertzJ. Rodriguez-Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Inventiones Mathematicae, 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[18]

F. Rodriguez-HertzJ. Rodriguez-Hertz and R. Ures, A non-dynamically coherent example on $\mathbb{T}^3$, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.  Google Scholar

[19]

V. Rokhlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk., 22 (1967), 3-56.   Google Scholar

[20]

Y. Sinai, Simple random walks on tori, Journal of Statistical Physics, 94 (1999), 695-708.  doi: 10.1023/A:1004564824697.  Google Scholar

[21]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynamical Systems, 6 (1986), 449-473.  doi: 10.1017/S0143385700003606.  Google Scholar

[22]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75–165.  Google Scholar

[23]

O. Zeitouni, Random walks in random environment, In Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004,189–312. doi: 10.1007/978-3-540-39874-5_2.  Google Scholar

show all references

References:
[1]

D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[2]

A. AvilaM. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows, Journal of the European Mathematical Society, 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.  Google Scholar

[3]

A. Candel and L. Conlon, Foliations I, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/gsm/023.  Google Scholar

[4]

J.-P. Conze and Y. Guivarc'h, Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique, Colloquium Mathematicum, 84 (2000), 457-480.  doi: 10.4064/cm-84/85-2-457-480.  Google Scholar

[5]

D. Dolgopyat, B. Fayad, and M. Saprykina, Erratic behavior for 1-dimensional Random Walks in a Liouville quasi-periodic environment., preprint, 2019, arXiv: 1901.10709. Google Scholar

[6]

M. Gorodin and B. Lifsic, Central limit theorem for stationary Markov processes, In Third Vilnius Conference on Probability and Statistics, volume 1, 1981,147–148. Google Scholar

[7] B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University Press, New York, 2003.  doi: 10.1017/CBO9780511998188.  Google Scholar
[8]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A Survey of Partially Hyperbolic Dynamics, In Partially Hyperbolic Dynamics, Laminations and Teichmüller Flow, Fields Institute Communications, vol. 51, 2007, 35–88.  Google Scholar

[9]

M. W. Hirsch, C. C. Pugh, and M. I. Shub, Invariant Manifolds, Springer Berlin Heidelberg, 1977. Google Scholar

[10]

V. Kaloshin and Y. Sinai, Simple random walks along orbits of Anosov diffeomorphisms, Tr. Mat. Inst. Steklova, 228 (2000), 236-245.   Google Scholar

[11]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210.  doi: 10.4310/MRL.1996.v3.n2.a6.  Google Scholar

[12]

Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

[13]

C. Kipnis and S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications Math. Physics, 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[14]

J. Neveu and A. Feinstein, Mathematical Foundations of the Calculus of Probability, Holden-Day, Inc. San Francisco, Calif.-London-Amsterdam, 1965.  Google Scholar

[15]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, Zürich, 2004. doi: 10.4171/003.  Google Scholar

[16]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS), 2 (2000), 1-52.  doi: 10.1007/s100970050013.  Google Scholar

[17]

F. Rodriguez-HertzJ. Rodriguez-Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Inventiones Mathematicae, 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[18]

F. Rodriguez-HertzJ. Rodriguez-Hertz and R. Ures, A non-dynamically coherent example on $\mathbb{T}^3$, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.  Google Scholar

[19]

V. Rokhlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk., 22 (1967), 3-56.   Google Scholar

[20]

Y. Sinai, Simple random walks on tori, Journal of Statistical Physics, 94 (1999), 695-708.  doi: 10.1023/A:1004564824697.  Google Scholar

[21]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynamical Systems, 6 (1986), 449-473.  doi: 10.1017/S0143385700003606.  Google Scholar

[22]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75–165.  Google Scholar

[23]

O. Zeitouni, Random walks in random environment, In Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004,189–312. doi: 10.1007/978-3-540-39874-5_2.  Google Scholar

[1]

Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058

[2]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519

[3]

Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517

[4]

Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3199-3215. doi: 10.3934/dcdsb.2020058

[5]

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

[6]

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

[7]

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

[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234

[9]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

[10]

Nguyen Huu Du, Nguyen Hai Dang. Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2693-2712. doi: 10.3934/cpaa.2014.13.2693

[11]

Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233

[12]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100

[13]

Veronika Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences & Engineering, 2015, 12 (2) : 393-413. doi: 10.3934/mbe.2015.12.393

[14]

Tom Goldstein, Xavier Bresson, Stan Osher. Global minimization of Markov random fields with applications to optical flow. Inverse Problems & Imaging, 2012, 6 (4) : 623-644. doi: 10.3934/ipi.2012.6.623

[15]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785

[16]

Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286

[17]

Jérôme Coville, Nicolas Dirr, Stephan Luckhaus. Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks & Heterogeneous Media, 2010, 5 (4) : 745-763. doi: 10.3934/nhm.2010.5.745

[18]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

[19]

Jiuping Xu, Pei Wei. Production-distribution planning of construction supply chain management under fuzzy random environment for large-scale construction projects. Journal of Industrial & Management Optimization, 2013, 9 (1) : 31-56. doi: 10.3934/jimo.2013.9.31

[20]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (134)
  • HTML views (152)
  • Cited by (0)

Other articles
by authors

[Back to Top]