Large deviations in dynamical systems and stochastic processes
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- by Yuri Kifer
- Trans. Amer. Math. Soc. 321 (1990), 505-524
- DOI: https://doi.org/10.1090/S0002-9947-1990-1025756-7
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Abstract:
The paper exhibits a unified approach to large deviations of dynamical systems and stochastic processes based on the existence of a pressure functional and on the uniqueness of equilibrium states for certain dense sets of functions. This enables us to generalize recent results from [OP, Y, and D] on large deviations for dynamical systems, as well, as to recover Donsker-Varadhan’s [DV2] large deviation estimates for Markov processes.References
- A. de Acosta, Upper bounds for large deviations of dependent random vectors, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 4, 551–565. MR 791911, DOI 10.1007/BF00532666
- A. de Acosta, Large deviations for vector-valued functionals of a Markov chain: lower bounds, Ann. Probab. 16 (1988), no. 3, 925–960. MR 942748
- 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. MR 749753
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
- 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. MR 912374, DOI 10.1017/S0143385700004107
- Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848 M. Denker, Large deviations and the pressure function, Preprint.
- Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675 N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
- 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. MR 361998, DOI 10.1073/pnas.72.3.780
- 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. MR 386024, DOI 10.1002/cpa.3160280102
- 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. MR 425380, DOI 10.1002/cpa.3160290606
- Richard S. Ellis, Large deviations for a general class of random vectors, Ann. Probab. 12 (1984), no. 1, 1–12. MR 723726
- Jurgen Gertner, On large deviations from an invariant measure, Teor. Verojatnost. i Primenen. 22 (1977), no. 1, 27–42 (Russian, with German summary). MR 0471040
- Yuri Kifer, Random perturbations of dynamical systems, Progress in Probability and Statistics, vol. 16, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 1015933, DOI 10.1007/978-1-4615-8181-9
- Yuri Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel J. Math. 70 (1990), no. 1, 1–47. MR 1057266, DOI 10.1007/BF02807217
- M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
- Artur O. Lopes, Entropy and large deviation, Nonlinearity 3 (1990), no. 2, 527–546. MR 1054587
- 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. MR 935534, DOI 10.1090/S0002-9947-1989-0935534-4
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- 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. MR 511655
- Y\B{o}ichir\B{o} 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. MR 780769, DOI 10.1016/S0924-6509(08)70404-5
- 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. MR 1009507, DOI 10.1007/BF01044443
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
- Lai-Sang Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525–543. MR 975689, DOI 10.1090/S0002-9947-1990-0975689-7
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 321 (1990), 505-524
- MSC: Primary 60F10; Secondary 28D20, 58F15
- DOI: https://doi.org/10.1090/S0002-9947-1990-1025756-7
- MathSciNet review: 1025756