Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Lévy processes in semisimple Lie groups
and stability of stochastic flows


Author: Ming Liao
Journal: Trans. Amer. Math. Soc. 350 (1998), 501-522
MSC (1991): Primary 58G32; Secondary 60H10
DOI: https://doi.org/10.1090/S0002-9947-98-01730-9
MathSciNet review: 1373644
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the asymptotic stability of stochastic flows on compact spaces induced by Levy processes in semisimple Lie groups. It is shown that the Lyapunov exponents can be determined naturally in terms of root structure of the Lie group and there is an open subset whose complement has a positive codimension such that, after a random rotation, each of its connected components is shrunk to a single moving point exponentially under the flow.


References [Enhancements On Off] (What's this?)

  • 1. Applebaum, D. and Kunita, H., ``Lévy flows on manifolds and Lévy processes on Lie groups", J. Math. Kyoto Univ. 33-4, pp 1105-1125 (1993) MR 95d:58140
  • 2. Baxendale, P.H., ``Asymptotic behaviour of stochastic flows of diffeomorphisms: two case studies", Probab. Th. Rel. Fields 73, pp 51-85 (1986) MR 88c:58073
  • 3. Carverhill, A.P., ``Flows of stochastic dynamical systems: ergodic theory", Stochastics 14, pp 273-317 (1985) MR 87c:58059
  • 4. Elworthy, K.D., ``Geometric aspects of diffusions on manifolds", (Ecole d'Eté de Probabilités de Saint Flour XVII, July 1987), Lect Notes in Math 1362, pp 276-425 (1989) MR 90c:58187
  • 5. Guivarc'h, Y. et Raugi, A., ``Frontière de Furstenberg, propriétés de contraction et convergence", Z. Wahr verw Gebiete 68, pp 187-242 (1985) MR 86h:60126
  • 6. Helgason, S., ``Differential geometry, Lie groups, and symmetric spaces", Academic Press (1978) MR 80k:53081
  • 7. Hunt, G.A., ``Semigroup of measures on Lie groups", Transactions AMS 81 (2), pp 264-293 (1956) MR 18:54a
  • 8. Ikeda, N. and Watanabe, S., ``Stochastic differential equations and diffusion processes", Second ed, North-Holland (1989) MR 90m:60069
  • 9. Liao, M., ``Stochastic flows on the boundaries of Lie groups", Stochastics and Stochastic Reports, vol 39, pp 213-237 (1992) MR 95d:58143
  • 10. Liao, M., ``Stochastic flows on the boundaries of SL(n,R)", Probab Theo & Rel Fields 96, pp 261-281 (1993) MR 95d:58144
  • 11. Liao, M., ``Invariant diffusion processes in Lie groups and stochastic flows", Proceedings of 1993 Summer Research Institute on Stochastic Analysis, July 1993, Cornell Univ (ed M. Cranston & M. Pinsky), Proc. Sympos. Pure Math., vol. 57, Amer. Math. Soc., Providence, RI, 1995, pp. 575-591. MR 96d:58154
  • 12. Malliavin, M.P. & Malliavin, P., ``Factorisations et lois limites de la diffusion horizontale au-dessus d'un espace Riemannien symmetrique", Lecture Notes in Math. 404, Springer-Verlag, pp 164-217 (1974) MR 50:11478

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58G32, 60H10

Retrieve articles in all journals with MSC (1991): 58G32, 60H10


Additional Information

Ming Liao
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email: liaomin@mail.auburn.edu

DOI: https://doi.org/10.1090/S0002-9947-98-01730-9
Keywords: Levy processes, semisimple Lie groups, stochastic flows
Received by editor(s): June 19, 1995
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society