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Large deviations and stochastic flows of diffeomorphisms. (English) Zbl 0638.60035
Previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold M are extended and then applied to the analysis of the Lyapunov exponents of a stochastic flow of diffeomorphisms of M. An approximation argument relates these results to the behavior near the diagonal $$\Delta$$ in M 2 of the associated two-point motion.
Finally it is shown, under appropriate non-degeneracy conditions, that the two-point motion is ergodic on M 2-$$\Delta$$ if the top Lyapunov exponent is positive.

##### MSC:
 60F10 Large deviations 58J65 Diffusion processes and stochastic analysis on manifolds 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes
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##### References:
 [1] Arnold, L., Kliemann, W.: Large deviations of linear stochastic differential equations. In: Engelbert, H.J. Schmidt (ed.) Proceedings of the Fifth IFIP Working Conference on Stochastic Differential Systems, Eisenach 1986 (Lect. Notes Control Inf. Sci. Vol. 96) Berlin Heidelberg New York: Springer 1987 · Zbl 0632.60021 [2] Arnold, L., Oeljeklaus, E., Pardoux, E.: Almost sure and moment stability for linear Itô equations. In: Arnold, L., Wihstutz, V. (eds.) Lyapunov exponents (Lect. Notes Math., Vol. 1186, pp. 129–159) Berlin Heidelberg New York: Springer 1986 · Zbl 0588.60049 [3] Baxendale, P.: The Lyapunov spectrum of a stochastic flow of diffeomorphisms. In: Arnold, L., Wihstutz, V. (eds.) Lyapunov exponents. (Lect. Notes Math., Vol. 1186, pp. 322–337) Berlin Heidelberg New York: Springer 1986 · Zbl 0592.60047 [4] Baxendale, P.: Moment stability and large deviations for linear stochastic differential equations. In: Ikeda, N. (ed) Proceedings of the Taniguchi Symposium on Probabilistic Methods in Mathematical Physics. Katata and Kyoto 1985, pp. 31–54. Tokyo: Kinokuniya 1987 [5] Baxendale, P.: Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. Probab. Theor. Rel. Fields., in press · Zbl 0645.58043 [6] Bony, J.-M.: Principe du maximum, unicité du problem de Cauchy et inéquality de Harnack pour les operateurs elliptiques dégeneres. Ann. Inst. Fourier 19, 277–304 (1969) [7] Carverhill, A.: Flows of stochastic dynamical systems: ergodic theory. Stochastics 14, 273–317 (1985) · Zbl 0536.58019 [8] Carverhill, A.: A formula for the Lyapunov numbers of a stochastic flow. Applications to a perturbation theorem. Stochastics 14, 209–226 (1985) · Zbl 0557.60048 [9] Chappell, M.J.: Bounds for average Lyapunov exponents of gradient stochastic systems. In: Arnold, L., Wihstutz, V. (eds.) Lyapunov exponents. (Lect. Notes Math., Vol. 1186, pp. 308–321) Berlin Heidelberg New York: Springer 1986 · Zbl 0591.58036 [10] Cheeger, J., Ebin, D.G.: Comparison theorems in Riemanian geometry. Amsterdam New York: North Holland/American Elsevier 1975 · Zbl 0309.53035 [11] Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure. Appl. Math. 28, 1–47 (1975) · Zbl 0323.60069 · doi:10.1002/cpa.3160280102 [12] Elworthy, K.D., Stroock, D.W.: Large deviation theory for mean exponents of stochastic flows. In: Albeverio, S., Blanchard, P., Streit, L. (eds.) Stochastic processes, mathematics and physics (Lect. Notes Math., Vol. 1158, pp. 72–80) Berlin Heidelberg New York: 1986 [13] Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963) · Zbl 0203.19102 · doi:10.1090/S0002-9947-1963-0163345-0 [14] Kato, T.: Perturbation theory of linear operators, Berlin Heidelberg New York: Springer 1966 · Zbl 0148.12601 [15] Khas’menskii, R.K.: Ergodic properties and stabilization of the solution to the Cauchy problem for parabolic equations. Theor. Probab. Appl. 5, 179–196 (1960) · Zbl 0106.12001 · doi:10.1137/1105016 [16] Kusuoka, S., Stroock, D.W.: Applications of the Malliavin calculus, I. In: Itô, K. (ed.) Stochastic analysis, pp. 271–307. Proceedings, Taniguchi Int. Symp. at Katata and Kyoto in 1982. Tokyo Amsterdam: Kinokuniya/North Holland 1984 [17] Kusuoka, S., Stroock, D.W.: Applications of the Malliavin calculus, II. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 32, 1–76 (1985) · Zbl 0568.60059 [18] Ledrappier, F., Young, L.-S.: Entropy formula for random transformations. Probab. Theor. Rel. Fields 80, 217–240 (1989) · Zbl 0638.60054 · doi:10.1007/BF00356103 [19] Pinsky, R.: On evaluating the Donsker-Varadhan I-functional. Ann. Probab. 13, 342–362 (1985) · Zbl 0607.60024 · doi:10.1214/aop/1176992995 [20] Stroock, D.W.: An introduction of the theory of large deviations. (Universitext Series) New York Heidelberg Berlin: Springer 1984 · Zbl 0552.60022 [21] Stroock, D.W., On the rate at which a homogeneous diffusion approaches a limit, an application of the large deviation theory of certain stochastic integrals. Ann. Probab. 14, 840–859 (1986) · Zbl 0604.60076 · doi:10.1214/aop/1176992441
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