Previous |  Up |  Next


Title: Lyapunov exponents for stochastic differential equations on semi-simple Lie groups (English)
Author: Ruffino, Paulo R. C.
Author: San Martin, Luiz A. B.
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 37
Issue: 3
Year: 2001
Pages: 207-231
Summary lang: English
Category: math
Summary: 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. (English)
Keyword: Lyapunov exponents
Keyword: stochastic differential equations
Keyword: semi-simple Lie groups
Keyword: flag manifolds
MSC: 22E46
MSC: 58J65
MSC: 60H10
idZBL: Zbl 1090.60054
idMR: MR1860184
Date available: 2008-06-06T22:28:54Z
Last updated: 2012-05-10
Stable URL:
Reference: [1] 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. Zbl 0588.60047, MR 0850072
Reference: [2] 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. Zbl 0588.60049, MR 0850074
Reference: [3] Arnold L., Imkeller P.: Furstenberg-Khasminskii formulas for Lyapunov exponents via antecipative calculus.Stochastics and Stochastics Reports, 54 (1+2) (1995), 127–168. MR 1382281
Reference: [4] Baxendale P. H.: Asymptotic behavior of stochastic flows of diffeomorphisms: Two case studies.Probab. Theory Related Fields, 73 (1986), 51–85. MR 0849065
Reference: [5] 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. Zbl 0592.60047, MR 0850087
Reference: [6] Borel A.: Kählerian coset spaces of semi-simple Lie groups.Proc. Nat. Acad. Sci. 40 (1954), 1147–1151. MR 0077878
Reference: [7] Carverhill A. P.: Flows of stochastic dynamical systems: Ergodic Theory.Stochastics 14 (1985), 273–317. Zbl 0536.58019, MR 0805125
Reference: [8] Carverhill A. P.: A Formula for the Lyapunov numbers of a stochastic flow. Application to a perturbation theorem.Stochastics 14 (1985), 209–226. Zbl 0557.60048, MR 0800244
Reference: [9] Carverhill A. P.: A non-random Lyapunov spectrum for non-linear stochastic systems.Stochastics 17 (1986), 253–287. MR 0854649
Reference: [10] Carverhill A. P., Elworthy K. D.: Lyapunov exponents for a stochastic analogue of the geodesic flow.Trans. Amer. Math. Soc. 295 (1986), 85–105. Zbl 0593.58048, MR 0831190
Reference: [11] 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. Zbl 0524.43008, MR 0707179
Reference: [12] Furstenberg H., Kesten H.: Products of random matrices.Ann. Math. Stat. 31 (1960), 457–469. Zbl 0137.35501, MR 0121828
Reference: [13] Guivarc’h Y., Raugi A.: Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence.Z. Wahrscheinlinchkeitstheor. Verw. Geb. 69 (1985), 187–242. Zbl 0558.60009, MR 0779457
Reference: [14] Helgason S.: Differential geometry, Lie groups and symmetric spaces.Academic Press (1978). Zbl 0451.53038, MR 0514561
Reference: [15] Ikeda N., Watanabe S.: Stochastic differential equations and diffusion processes.North-Holland (1981). Zbl 0495.60005, MR 1011252
Reference: [16] Khashminskii R. Z.: Stochastic stability of differential equations.Sijthoff and Noordhoff, Alphen (1980). MR 0600653
Reference: [17] Kobayashi S., Nomizu K.: Foundations of differential geometry.Interscience Publishers (1963 and 1969). Zbl 0119.37502, MR 0152974
Reference: [18] Liao M.: Stochastic flows on the boundaries of Lie groups.Stochastics Stochastics Rep. 39 (1992), 213–237. Zbl 0754.60016, MR 1275123
Reference: [19] Liao M.: Liapunov Exponents of Stochastic Flows.Ann. Probab. 25 (1997), 1241–1256. MR 1457618
Reference: [20] Liao M.: Invariant diffusion processes in Lie groups and stochastic flows.Proc. of Symposia in Pure Math. 57 (1995), 575–591. Zbl 0839.58065, MR 1335499
Reference: [21] 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. MR 0359023
Reference: [22] Oseledec V. I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc. 19 (1968), 197–231. MR 0240280
Reference: [23] Ruelle D.: Ergodic theory of differentiable dynamical systems.I.H.E.S. – Publ. Math. 50, (1979), 275–306. Zbl 0426.58014, MR 0556581
Reference: [24] 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. Zbl 0641.93069, MR 0885003
Reference: [25] Sussmann H., Jurdjevic V.: Controllability of nonlinear systems.J. Differential Equations 12 (1972), 95–116. MR 0338882
Reference: [26] 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. MR 0954647
Reference: [27] Warner G.: Harmonic Analysis on Semi-simple Lie Groups.Springer-Verlag (1972). Zbl 0265.22021


Files Size Format View
ArchMathRetro_037-2001-3_4.pdf 462.6Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo