| 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:
|
http://hdl.handle.net/10338.dmlcz/107799 |
| . |
| 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 |
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