Lyapunov exponents for stochastic differential equations on semi-simple Lie groups
Ruffino, Paulo R. C. ; San Martin, Luiz A. B.
Archivum Mathematicum, Tome 037 (2001), p. 207-231 / Harvested from Czech Digital Mathematics Library
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With an intrinsic approach on semi-simple Lie groups we find a Furstenberg–Khasminskii type formula for the limit of the diagonal component in the Iwasawa decomposition. It is an integral formula with respect to the invariant measure in the maximal flag manifold of the group (i.e. the Furstenberg boundary $B=G/MAN$). Its integrand involves the Borel type Riemannian metric in the flag manifolds. When applied to linear stochastic systems which generate a semi-simple group the formula provides a diagonal matrix whose entries are the Lyapunov spectrum. Some Brownian motions on homogeneous spaces are discussed.

Publié le : 2001-01-01
Classification:  22E46,  58J65,  60H10 
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     author = {Paulo R. C. Ruffino and Luiz A. B. San Martin},
     title = {Lyapunov exponents for stochastic differential equations on semi-simple Lie groups},
     journal = {Archivum Mathematicum},
     volume = {037},
     year = {2001},
     pages = {207-231},
     zbl = {1090.60054},
     mrnumber = {1860184},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107799}
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Ruffino, Paulo R. C.; San Martin, Luiz A. B. Lyapunov exponents for stochastic differential equations on semi-simple Lie groups. Archivum Mathematicum, Tome 037 (2001) pp. 207-231. http://gdmltest.u-ga.fr/item/107799/

Arnold L.; Kliemann W.; Oeljeklaus E. Lyapunov exponents of linear stochastic systems, In Lyapunov Exponents (Eds. L. Arnold and W. Wihstutz), Lecture Notes Math. - Springer 1186 (1986), 85–128. (1986) | MR 0850072 | Zbl 0588.60047

Arnold L.; Oeljeklaus E.; Pardoux E. Almost sure and moment stability for linear Itô equations, In Lyapunov Exponents (Eds. L. Arnold and W. Wihstutz), Lecture Notes Math. - Springer 1186 (1986), 129–159. (1986) | MR 0850074 | Zbl 0588.60049

Arnold L.; Imkeller P. Furstenberg-Khasminskii formulas for Lyapunov exponents via antecipative calculus, Stochastics and Stochastics Reports, 54 (1+2) (1995), 127–168. (1995) | MR 1382281

Baxendale P. H. Asymptotic behavior of stochastic flows of diffeomorphisms: Two case studies, Probab. Theory Related Fields, 73 (1986), 51–85. (1986) | MR 0849065

Baxendale P. H. The Lyapunov spectrum of a stochastic flow of diffeomorphisms, In Lyapunov Exponents (Eds. L. Arnold and W. Wihstutz), Lecture Notes Math. - Springer 1186 (1986), 322–337. (1986) | MR 0850087 | Zbl 0592.60047

Borel A. Kählerian coset spaces of semi-simple Lie groups, Proc. Nat. Acad. Sci. 40 (1954), 1147–1151. (1954) | MR 0077878

Carverhill A. P. Flows of stochastic dynamical systems: Ergodic Theory, Stochastics 14 (1985), 273–317. (1985) | MR 0805125 | Zbl 0536.58019

Carverhill A. P. A Formula for the Lyapunov numbers of a stochastic flow. Application to a perturbation theorem, Stochastics 14 (1985), 209–226. (1985) | MR 0800244 | Zbl 0557.60048

Carverhill A. P. A non-random Lyapunov spectrum for non-linear stochastic systems, Stochastics 17 (1986), 253–287. (1986) | MR 0854649

Carverhill A. P.; Elworthy K. D. Lyapunov exponents for a stochastic analogue of the geodesic flow, Trans. Amer. Math. Soc. 295 (1986), 85–105. (1986) | MR 0831190 | Zbl 0593.58048

Duistermaat J. J.; Kolk J. A. C.; Varadarajan V. Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compositio Math. 49 (1983), 309–398. (1983) | MR 0707179 | Zbl 0524.43008

Furstenberg H.; Kesten H. Products of random matrices, Ann. Math. Stat. 31 (1960), 457–469. (1960) | MR 0121828 | Zbl 0137.35501

Guivarc’H Y.; Raugi A. Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrscheinlinchkeitstheor. Verw. Geb. 69 (1985), 187–242. (1985) | MR 0779457 | Zbl 0558.60009

Helgason S. Differential geometry, Lie groups and symmetric spaces, Academic Press (1978). (1978) | MR 0514561 | Zbl 0451.53038

Ikeda N.; Watanabe S. Stochastic differential equations and diffusion processes, North-Holland (1981). (1981) | MR 1011252 | Zbl 0495.60005

Khashminskii R. Z. Stochastic stability of differential equations, Sijthoff and Noordhoff, Alphen (1980). (1980) | MR 0600653

Kobayashi S.; Nomizu K. Foundations of differential geometry, Interscience Publishers (1963 and 1969). (1963) | MR 0152974 | Zbl 0119.37502

Liao M. Stochastic flows on the boundaries of Lie groups, Stochastics Stochastics Rep. 39 (1992), 213–237. (1992) | MR 1275123 | Zbl 0754.60016

Liao M. Liapunov Exponents of Stochastic Flows, Ann. Probab. 25 (1997), 1241–1256. (1997) | MR 1457618

Liao M. Invariant diffusion processes in Lie groups and stochastic flows, Proc. of Symposia in Pure Math. 57 (1995), 575–591. (1995) | MR 1335499 | Zbl 0839.58065

Malliavin M. P.; Malliavin P. Factorisations et lois limites de la diffusion horizontale au-dessus d’un espace Riemannien symmetrique, Lecture Notes Math. 404 (1974), 164–217. (1974) | MR 0359023

Oseledec V. I. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–231. (1968) | MR 0240280

Ruelle D. Ergodic theory of differentiable dynamical systems, I.H.E.S. – Publ. Math. 50, (1979), 275–306. (1979) | MR 0556581 | Zbl 0426.58014

San Martin L. A. B.; Arnold L. A Control problem related to the Lyapunov spectrum of stochastic flows, Mat. Apl. Comput. 5 (1986), 31–64. (1986) | MR 0885003 | Zbl 0641.93069

Sussmann H.; Jurdjevic V. Controllability of nonlinear systems, J. Differential Equations 12 (1972), 95–116. (1972) | MR 0338882

Taylor J. C. The Iwasawa decomposition and the limiting behavior of Brownian motion on a symmetric space of non-compact type, Contemp. Math. AMS 73 (1988), 303–302. (1988) | MR 0954647

Warner G. Harmonic Analysis on Semi-simple Lie Groups, Springer-Verlag (1972). (1972) | Zbl 0265.22021