Comparaison des exposants de Lyapounov des processus markoviens multiplicatifs

Bougerol, Philippe

Annales de l'I.H.P. Probabilités et statistiques, Tome 24 (1988), p. 439-489 / Harvested from Numdam

Détail

Publié le : 1988-01-01

MR 978021 | Zbl 0672.60034

Bibliographie

[1] L. Arnold et V. Wihstutz éd., Lyapunov Exponents, Proceedings, Bremen, 1984. Lecture Notes in Math, n° 1186, 1986, Springer Verlag, Berlin, Heidelberg, New York. | MR 850066 | Zbl 0599.60061

[2] L. Arnold, W. Kliemann et E. Oejelklaus, Lyapunov Exponents of Linear Stochastic systems, in [1], 1986, p. 85-125. | MR 850072 | Zbl 0588.60047

[3] L. Arnold, E. Oejelklaus et E. Pardoux, Almost Sure and Moment Stability for Linear Ito Equations, in [1], 1986, p. 129-159. | MR 850074 | Zbl 0588.60049

[4] L. Arnold et L. San Martin, A Control Problem Related to the Lyapunov Spectrum of Stochastic Flows, Matematica Aplicada e computational, vol. 5, n° 1, 1986, p. 31-64. | MR 885003 | Zbl 0641.93069

[5] P.H. Baxendale, Lyapunov Exponents and Relative Entropy for a Stochastic Flow of Diffeomorphisms, Preprint, 1986. | MR 995809

[6] J.M. Bismut, Mécanique Aléatoire, Lecture Notes in Math., n° 866, Springer Verlag, Berlin, Heidelberg, New York. | MR 665595 | Zbl 0457.60002

[7] P. Bougerol et J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, 1985, Birkhäuser, P. P. M., Boston, Basel, Stuttgart. | MR 886674 | Zbl 0572.60001

[8] P. Bougerol, Limit Theorem for Products of Random Matrices with Markovian Dependence, Proceedings of the First International Congress of the Bernoulli Society, 1986 (à paraître). | MR 1092426 | Zbl 0682.60004

[9] P. Bougerol, Théorèmes limites pour les systèmes linéaires à coefficients markoviens, Probab. Th. Rel. Fields, vol. 78, 1988, p. 193-221. | MR 945109 | Zbl 0627.60054

[10] A. Caverhill, A Formula for the Lyapunov Numbers of a Stochastic Flow. Application to a Perturbation Theorem, Stochastics, vol. 14, 1985, p. 209-226. | MR 800244 | Zbl 0557.60048

[11] A. Caverhill, Furstenberg's Theorem for Nonlinear Stochastic Systems, Probab. Th. Rel. Fields, vol. 74, 1987, p. 529-534. | MR 876254 | Zbl 0592.60043

[12] A. Caverhill, Lyapunov Exponents for Stochastic Flows on Manifolds, in [1], 1986, p. 292-307. | MR 850085 | Zbl 0602.58053

[13] A. Caverhill, M. Chappell et K. Elworthy, Characteristic Exponents for Stochastic Flows, Stochastic Processes. Mathematics and Physics, S. ALBEVERIO et al. éd., Lecture Notes in Math, n° 1158, 1987, Springer Verlag, Berlin, Heidelberg, New York, p. 52-72. | MR 838558 | Zbl 0651.60036

[14] P. Chassaing, Convergence d'un produit semi markovien de matrices et simplicité du spectre de la limite, Preprint, 1986.

[15] C. Dellacherie et P.A. Meyer, Probabilités et Potentiels, tome 1, 1975, Hermann, Paris. | MR 488194 | Zbl 0323.60039

[16] F. Delyon, Y. Levy et B. Souillard, Anderson Localization for One- and Quasi One-Dimensional systems, J. Stat. Phys., vol. 41, 1985, p. 375. | MR 814838

[17] F. Delyon, B. Simon et B. Souillard, Localization for Off-diagonal Disorder and for Continuous Schrôdinger Operators, Commun. Math. Phys., vol. 109, 1987, p. 157- 165. | MR 879035 | Zbl 0634.35070

[18] N. Dunford et J. Schwartz, Linear Operators, vol. 1, 1958, Interscience Publishers, Wiley, New York. | MR 117523 | Zbl 0084.10402

[19] K. Elworthy, Stochastic Differential Equations on Manifolds, L. M. S. Lecture notes, 1982, Cambridge University Press. | MR 675100 | Zbl 0514.58001

[20] I.J. Goldsheid et G.A. Margulis, La condition de simplicité du spectre des exposants de Lyapounov, 1987 (à paraître).

[21] Y. Guivarc'H, Exposants caractéristiques des produits de matrices aléatoires en dépendance markovienne, in Probability measures on group 7, H. HEYER éd., Lecture Notes in Math, n° 1064, 1984, Springer Verlag, Berlin, Heidelberg, New York, p. 161-181. | MR 772409 | Zbl 0561.60014

[22] Y. Guivarc'H et A. Raugi, Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Zeit. für Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 69, 1985, p. 187-242. | MR 779457 | Zbl 0558.60009

[23] Y. Guivarc'H et A. Raugi. Quelques remarques sur les produits de matrices aléatoires indépendantes, C. R. Acad. Sci. Paris, t. 304, série I, 1987, p. 199-208. | MR 880926 | Zbl 0619.22010

[24] R.Z. Hashminski, Stochastic Stability of Differential Equations, Alphen, Sijthoff and Noordhoff, 1980. | Zbl 0441.60060

[25] S. Helgason, Differential Geometry, Lie groups, and Symmetric spaces, Academic press, New York, San Francisco, London, 1978. | MR 514561 | Zbl 0451.53038

[26] K. Hewitt et A. Ross, Abstract Harmonic Analysis 1, Springer Verlag, Berlin, Heidelberg, New York, 1963.

[27] K. Ichihara et H. Kunita, A Classification of the Second Order Degenerate Elliptic Operators and its Probabilistic Characterization, Zeit. für Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 30, 1974, p. 235-254. | MR 381007 | Zbl 0326.60097

[28] N. Ikeda et S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland-Kodansha, Amsterdam, Tokyo, 1981. | MR 637061 | Zbl 0495.60005

[29] I. Iscoe, P. Ney et E. Nummelin, Large Deviations of Uniformly Recurrent Markov Additive Processes, Adv. in Appl. Proba., vol. 6, p. 373-412. | MR 826590 | Zbl 0602.60034

[30] V. Jurdjevic et H. Sussmann, Control Systems on Lie Groups, J. Diff. Equs., vol. 12, 1972, p. 313-329. | MR 331185 | Zbl 0237.93027

[31] Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser, P. P. M., Boston, Basel, Stuttgart, 1986. | MR 884892 | Zbl 0604.28014

[32] J.F.C. Kingman, Subadditive Ergodic Theory, Ann. Proba., vol. 1, 1973, p. 883-909. | MR 356192 | Zbl 0311.60018

[33] S. Kobayashi et K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, Wiley, New York, 1969. | MR 238225 | Zbl 0175.48504

[34] S. Kotani, Limit Theorems of Hypoelliptic Diffusion Processes, in Probability Theory and Mathemetical Statistics, K. ITO et J. V. PROKHOROV éd., Lecture Notes in Math., n° 1021, 1983, Springer Verlag, Berlin, Heidelberg, New York, p. 333-338. | MR 736000 | Zbl 0525.60082

[35] H. Kunita, Supports of Diffusion Processes and Controllability Problems, Proceed. Intern. Symp. Stochastic Diff. Equs., Kyoto, 1976, p. 163-185, Wiley, New York, 1978. | MR 536011 | Zbl 0409.60063

[36] F. Ledrappier, Quelques Propriétés des Exposants Caractéristiques, in École d'été de Saint-Flour 12, 1982, P.L. Hennequin éd., Lecture Notes in Math., n° 1097, 1984, Springer Verlag, Berlin, Heidelberg, New York. | MR 876081 | Zbl 0578.60029

[37] F. Ledrappier, Positivity of the Exponent for Stationary Sequences of Matrices, in [1], 1986, p. 56-73. | MR 850070 | Zbl 0591.60036

[38] F. Ledrappier et K.L. Young, Entropy Formula for Random Transformations, Preprint, 1986.

[39] S.A. Molchanov, The Structure of Eigenfunctions of One-Dimensional Unordered Structures, Math. U.S.S.R. Izvestija, vol. 12, 1978, p. 69-101. | Zbl 0401.34023

[40] M. Obata, The Conjectures About Conformal Transformations, J. Diff. Geom., vol. 6, 1971, p. 247-258. | MR 303464 | Zbl 0236.53042

[41] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, London, 1967. | MR 226684 | Zbl 0153.19101

[42] D. Revuz, Markov Chains, North Holland, Amsterdam, New York, Oxford, 1984. | MR 758799 | Zbl 0539.60073

[43] G. Royer, Croissance exponentielle de produits markoviens de matrices aléatoires, Ann. I.H.P., vol. 16, 1980, p. 49-62. | Numdam | MR 575176 | Zbl 0433.60073

[44] D. Stroock et S.R.S. Varadhan, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle, Proc. 6th Berkeley Symp. Math. Stat. Prob. III, 1972, p. 333-368. | MR 400425 | Zbl 0255.60056

[45] H.J. Sussman, Lie Brackets, Real Analyticity and Geometric Control Theory, in Differential geometric control theory, Birkhâuser, Boston, Basel, Stuttgart, 1983, p. 1-176. | MR 708500 | Zbl 0545.93002

[46] A.D. Virtser, On the Simplicity of the Spectrum of the Lyapunov Characteristic Indices of a Product of Random Matrices, Theor. Proba. Appl., vol. 28, 1984, p. 122- 135. | Zbl 0529.60005