Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-11T20:27:33.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupling methods for random topological Markov chains

Published online by Cambridge University Press:  20 October 2015

MANUEL STADLBAUER
MANUEL STADLBAUER*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68530, CEP 21945-970, Rio de Janeiro (RJ), Brazil email manuel.stadlbauer@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We apply coupling techniques in order to prove that the transfer operators associated with random topological Markov chains and non-stationary shift spaces with the big images and preimages property have a spectral gap.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 3 , May 2017 , pp. 971 - 994
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnoux, P. and Fisher, A. M.. Anosov families, renormalization and non-stationary subshifts. Ergod. Th. & Dynam. Sys. 25 (2005), 661–709.CrossRefGoogle Scholar
Bogenschütz, T. and Gundlach, V. M.. Ruelle’s transfer operator for random subshifts of finite type. Ergod. Th. & Dynam. Sys. 15 (1995), 413–447.Google Scholar
Bressaud, X., Fernández, R. and Galves, A.. Decay of correlations for non-Hölderian dynamics: A coupling approach. Electron. J. Probab. 4 (1999), 19 (electronic).Google Scholar
Buzzi, J.. Exponential decay of correlations for random Lasota–Yorke maps. Comm. Math. Phys. 208 (1999), 25–54.Google Scholar
Crauel, H.. Random Probability Measures on Polish Spaces. Taylor & Francis, London, 2002.Google Scholar
Denker, M., Kifer, Y. and Stadlbauer, M.. Thermodynamic formalism for random countable Markov shifts. Discrete Contin. Dyn. Syst. 22 (2008), 131–164.Google Scholar
Doeblin, W. and Fortet, R.. Sur des chaînes à liaisons complètes. Bull. Soc. Math. France 65 (1937), 132–148.Google Scholar
Evstigneev, I. V. and Pirogov, S. A.. Stochastic nonlinear Perron–Frobenius theorem. Positivity 14 (2010), 43–57.Google Scholar
Fisher, A. M.. Nonstationary mixing and the unique ergodicity of adic transformations. Stoch. Dyn. 9 (2009), 335–391.Google Scholar
Galatolo, S. and Pacifico, M. J.. Lorenz-like flows: Exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence. Ergod. Th. & Dynam. Sys. 30 (2010), 1703–1737.Google Scholar
Hairer, M. and Mattingly, J. C.. Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36 (2008), 2050–2091.Google Scholar
Hennion, H.. Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118 (1993), 627–634.Google Scholar
Kifer, Y.. Thermodynamic formalism for random transformations revisited. Stoch. Dyn. 8 (2008), 77–102.CrossRefGoogle Scholar
Kifer, Y. and Liu, P.-D.. Random dynamics. Handbook of Dynamical Systems. Vol. 1B. Elsevier BV, Amsterdam, 2006.Google Scholar
Orey, S.. Markov chains with stochastically stationary transition probabilities. Ann. Probab. 19 (1991), 907–928.Google Scholar
Sarig, O. M.. Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131 (2003), 1751–1758.Google Scholar
Shen, W. and van Strien, S.. On stochastic stability of expanding circle maps with neutral fixed points. Dyn. Syst. 28 (2013), 423–452.CrossRefGoogle Scholar
Stadlbauer, M.. On random topological Markov chains with big images and preimages. Stoch. Dyn. 10 (2010), 77–95.Google Scholar
Stadlbauer, M.. Corrigendum and addendum to: On random topological Markov chains with big images and preimages. Stoch. Dyn. 14 (2014), 1492001, 3 pp.CrossRefGoogle Scholar
Tulcea, C. T. I. and Marinescu, G.. Théorie ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 52 (1950), 140–147.Google Scholar
Villani, C.. Optimal Transport. Springer, Berlin, 2009.Google Scholar
Zhang, X.. Stochastic Monge–Kantorovich problem and its duality. Stochastics 85 (2013), 71–84.Google Scholar