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Perron-Frobenius theorem, large deviations, and random perturbations in random environments. (English) Zbl 0863.60062

The author derives first a version of the Perron-Frobenius theorem for random positive operators together with the Doeblin condition type exponential convergence results for transition probabilities of Markov chains in random stationary environments. Next, he obtains a Donsker-Varadhan type variational formula for such Markov chains which yields both large deviations estimates and the corresponding results on random perturbations of random expanding transformations.
Reviewer: Y.Kifer

MSC:

60J05 Discrete-time Markov processes on general state spaces
47B80 Random linear operators
37-XX Dynamical systems and ergodic theory
60F10 Large deviations
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References:

[1] Bogenschutz, T.: Entropy, pressure, and a variational principle for random dynamical systems. Random & Comp. Dyn.1, 99–116 (1992)
[2] Bogenschutz, T., Gundlach, V.M.: Symbolic dynamics for expanding random dynamical systems. Random & Comp. Dyn.1, 219–227 (1992)
[3] Bogenschutz, T., Gundlach, V.M.: Ruelle’s transfer operator for random subshifts of finite type. Ergod. Th. & Dynam. Sys.15, 413–447 (1995) · Zbl 0842.58055
[4] Cogburn, R.: The ergodic theory of Markov chains in random environments. Z. Wahrsch. Verw. Gebiete66, 109–128 (1984) · Zbl 0525.60074 · doi:10.1007/BF00532799
[5] Cogburn, R.: On direct convergence and periodicity for transition probabilities of Markov chains in random environments. Ann. Probab.18, 642–654 (1990) · Zbl 0707.60057 · doi:10.1214/aop/1176990850
[6] Cogburn, R.: On the central limit theorem for Markov chains in random environments. Ann. Probab.19, 587–604 (1991) · Zbl 0733.60038 · doi:10.1214/aop/1176990442
[7] Dubins, L.E., Freedman, D.A.: Invariant probabilities for certain Markov processes. Ann. Math. Stat.37, 837–848 (1966) · Zbl 0147.16404 · doi:10.1214/aoms/1177699364
[8] Kifer, Y.: Random Perturbations of Dynamical Systems. Birkhäuser, Boston, 1988 · Zbl 0659.58003
[9] Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc.321, 505–524 (1990) · Zbl 0714.60019 · doi:10.1090/S0002-9947-1990-1025756-7
[10] Kifer, Y.: Principal eigenvalue, topological pressure, and stochastic stability of equilibrium states. Israel J. Math.70, 1–47 (1990) · Zbl 0732.58037 · doi:10.1007/BF02807217
[11] Kifer, Y.: A discrete-time version of the Wentzell-Freidlin theory. Ann. Probab.18, 1676–1692 (1990) · Zbl 0718.60022 · doi:10.1214/aop/1176990641
[12] Kifer, Y.: Equilibrium states for random expanding transformations. Random & Comp. Dyn.1, 1–31 (1992) · Zbl 0788.58019
[13] Kifer, Y.: A variational approach to the random diffeomorphisms type random perturbations of a hyperbolic diffeomorphism. In: Mathematical Physics, Vol. X K. Schmüdgen (eds.), pp. 334–340, Springer, 1992 · Zbl 0947.37503
[14] Kifer, Y.: Principal eigenvalues and equilibrium states corresponding to weakly coupled parabolic systems of PDE. J. D’Analise Math.59, 89–102 (1992) · Zbl 0803.35065 · doi:10.1007/BF02790219
[15] Kifer, Y.: Topics in Large Deviations and Random Perturbations. Department of Mathematics, University of North Carolina at Chapel Hill, 1993 · Zbl 0788.58019
[16] Kifer, Y.: Fractals via random iterated function systems and random geometric constructions. In: Fractal Geometry and Stochastics. C. Bandt, S. Graf, M. Zähle, (eds.), Progress in Probability. Birkhäuser, 1995 · Zbl 0866.60020
[17] Khanin, K., Kifer, Y.: Thermodynamic formalism for random transformations and statistical mechanics. Amer. Math. Soc. Transl. (2)171, 107–140 (1996) · Zbl 0841.58017
[18] Orey, S.: Markov chains with stochastically stationary transition probabilities. Ann. Probab.19, 907–928 (1991) · Zbl 0735.60040 · doi:10.1214/aop/1176990328
[19] Seppäläinen, T.: Large deviations for Markov chains with random transitions. Ann. Probab.22, 713–748 (1994) · Zbl 0809.60032 · doi:10.1214/aop/1176988727
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