Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T08:41:05.457Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of Markov chains and stable manifolds for random diffeomorphisms

Published online by Cambridge University Press:  19 September 2008

M. Brin and
Yu. Kifer
M. Brin
Affiliation:
University of Maryland, College Park, MD 20742, USA;
Yu. Kifer
Affiliation:
Hebrew University of Jerusalem, Jerusalem, Israel
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider the Markov chain on a compact manifold M generated by a sequence of random diffeomorphisms, i.e. a sequence of independent Diff2(M)-valued random variables with common distribution. Random diffeomorphisms appear for instance when diffusion processes are considered as solutions of stochastic differential equations. We discuss the global dynamics of Markov chains with continuous transition densities and construct non-random stable foliations for random diffeomorphisms.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 3 , September 1987, pp. 351 - 374
Copyright
Copyright © Cambridge University Press 1987

References

[BN]Brin, M. & Nitecki, Z.. Absolute continuity of stable foliations in Hilbert space. Preprint, 1987.Google Scholar
[C]Carverhill, A. P.. A nonrandom Lyapunov spectrum for non-linear stochastic dynamical systems. Stochastic 17 (1986), 253–287.CrossRefGoogle Scholar
[D]Doob, J. L.. Stochastic Processes. Wiley, New York, 1953.Google Scholar
[E]Elworthy, D.. Stochastic Differential Equations on Manifolds. London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1982.CrossRefGoogle Scholar
[FHY]Fathi, A.Herman, M. & Yoccoz, J. C.. Proof of Pesin's stable manifold theorem. In Geometric Dynamics (ed. J. Palis, Jr.), Lecture Notes in Math. 1007 (Springer, 1983), 177–215.CrossRefGoogle Scholar
[HS]Hewitt, E. & Stromberg, K.. Real and Abstract Analysis. Springer, New York, 1965.Google Scholar
[Ki]Kifer, Yu.. Ergodic Theory of Random Transformations. Birkhauser, 1986.CrossRefGoogle Scholar
[Ku]Kunita, H.. Stochastic differential equations and stochastic flows of iffeomorphisms. Lecture Notes in Math. 1097(Springer, 1984), 143–303.CrossRefGoogle Scholar
[L]Ledrappier, F.. Quelques proprietes des exposants characteristiques. Lecture Notes in Math. 1097 (Springer, 1984), 305–396.CrossRefGoogle Scholar
[Or]Orey, S.. Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold Co., London, 1971.Google Scholar
[Os]Oseledec, V. I.. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
[P]Pesin, Ya. B.. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR Izvestija 10 (1976), 1261–1305.CrossRefGoogle Scholar
[Re]Revuz, D.. Markov Chains. North-Holland Publishing Co., Amsterdam, 1975.Google Scholar
[Ro]Rosenblatt, M.. Markov Processes. Structure and Asymptotic Behavior. Springer, Berlin, 1971.CrossRefGoogle Scholar
[Ru]Ruelle, D.. Ergodic theory of diflerentiable dynamical systems. Publ. Math. IHES 50 (1979), 275–306.CrossRefGoogle Scholar
[Rul]Ruelle, D.. Characteristic exponents and invariant manifolds in Hilbert space. Annals of Math. 115 (1982), 243–290.CrossRefGoogle Scholar