×

zbMATH — the first resource for mathematics

Large deviations in dynamical systems and stochastic processes. (English) Zbl 0714.60019
The paper presents a general large deviations result for families of random probability measures on a compact metric space \(X\). An upper large deviation bound is established provided the pressure of every continuous function on \(X\) exists; the rate functional is the convex conjugate of the pressure. A lower bound is more difficult to obtain. The lower bound is proved assuming that there exist a countable dense set of continuous functions on \(X\) such that all finite linear combinations of functions from this set have a unique equilibrium state.
Reviewer: H.Crauel

MSC:
60F10 Large deviations
37H99 Random dynamical systems
37D99 Dynamical systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. de Acosta, Upper bounds for large deviations of dependent random vectors, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 4, 551 – 565. · Zbl 0547.60033 · doi:10.1007/BF00532666 · doi.org
[2] A. de Acosta, Large deviations for vector-valued functionals of a Markov chain: lower bounds, Ann. Probab. 16 (1988), no. 3, 925 – 960. · Zbl 0657.60037
[3] Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. · Zbl 0641.47066
[4] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. · Zbl 0308.28010
[5] M. Brin and Yu. Kifer, Dynamics of Markov chains and stable manifolds for random diffeomorphisms, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 351 – 374. · Zbl 0656.58033 · doi:10.1017/S0143385700004107 · doi.org
[6] Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181 – 202. · Zbl 0311.58010 · doi:10.1007/BF01389848 · doi.org
[7] M. Denker, Large deviations and the pressure function, Preprint. · Zbl 0769.60025
[8] Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. · Zbl 0328.28008
[9] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958. · Zbl 0084.10402
[10] Monroe D. Donsker and S. R. S. Varadhan, On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 780 – 783. · Zbl 0353.49039
[11] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. I. II, Comm. Pure Appl. Math. 28 (1975), 1 – 47; ibid. 28 (1975), 279 – 301. · Zbl 0323.60069 · doi:10.1002/cpa.3160280102 · doi.org
[12] M. D. Donsker and S. R. S. Varadhan, On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math. 29 (1976), no. 6, 595 – 621. · Zbl 0356.35065 · doi:10.1002/cpa.3160290606 · doi.org
[13] Richard S. Ellis, Large deviations for a general class of random vectors, Ann. Probab. 12 (1984), no. 1, 1 – 12. · Zbl 0534.60026
[14] Jurgen Gertner, On large deviations from an invariant measure, Teor. Verojatnost. i Primenen. 22 (1977), no. 1, 27 – 42 (Russian, with German summary).
[15] Yuri Kifer, Random perturbations of dynamical systems, Progress in Probability and Statistics, vol. 16, Birkhäuser Boston, Inc., Boston, MA, 1988. · Zbl 0659.58003
[16] Yuri Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel J. Math. 70 (1990), no. 1, 1 – 47. · Zbl 0732.58037 · doi:10.1007/BF02807217 · doi.org
[17] M. A. Krasnosel\(^{\prime}\)skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964.
[18] Artur O. Lopes, Entropy and large deviation, Nonlinearity 3 (1990), no. 2, 527 – 546. · Zbl 0703.58032
[19] Steven Orey and Stephan Pelikan, Deviations of trajectory averages and the defect in Pesin’s formula for Anosov diffeomorphisms, Trans. Amer. Math. Soc. 315 (1989), no. 2, 741 – 753. · Zbl 0691.60022
[20] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. · Zbl 0193.18401
[21] David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. · Zbl 0401.28016
[22] Yōichirō Takahashi, Entropy functional (free energy) for dynamical systems and their random perturbations, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 437 – 467. · doi:10.1016/S0924-6509(08)70404-5 · doi.org
[23] S. Vaienti, Computing the pressure for Axiom-A attractors by time series and large deviations for the Lyapunov exponent, J. Statist. Phys. 56 (1989), no. 3-4, 403 – 413. · Zbl 0712.58040 · doi:10.1007/BF01044443 · doi.org
[24] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. · Zbl 0475.28009
[25] Lai-Sang Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525 – 543. · Zbl 0721.58030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.