Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Stochastic flows on the boundaries of SL(n, R)
Download PDF
Download PDF
  • Published: June 1993

Stochastic flows on the boundaries of SL(n, R)

  • Ming Liao1 

Probability Theory and Related Fields volume 96, pages 261–281 (1993)Cite this article

  • 88 Accesses

  • 3 Citations

  • Metrics details

Summary

We study the asymptotic stability of the stochastic flows on a class of compact spaces induced by a diffusion process in SL(n, R) or GL(n, R). These compact spaces are called boundaries of SL(n, R), which include SO(n), the flag manifold, the sphereS n−1 and the Grassmannians. The one point motions of these flows are Brownian motions. For almost every, ω, we determine the set of stable points. This is a random open set whose complement has zero Lebesgue measure. The distance between any two points in the same component of this set tends to zero exponentially fast under the flow. The Lyapunov exponents at stable points are computed explicitly. We apply our results to a stochastic flow onS n−2 generated by a stochastic differential equation which exhibits some nice symmetry.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Baxendale, P.H.: Asymptotic behaviour of stochastic flows of diffeomorphisms: two case studies. Probab. Theory Relat. Fields73, 51–85 (1986)

    Google Scholar 

  2. Dynkin, E.B.: Non-negative eigenfunctions of the Laplace-Betrami operators and Brownian motion in certain symmetric spaces. Dokl. Akad. Nauk. SSSR141, 1433–1436 (1961)

    Google Scholar 

  3. Furstenberg, H.: A Poisson formula for semisimple Lie groups. Ann Math77 (no. 2), 335–386 (1962)

    Google Scholar 

  4. Helgason, S.: Group and geometric analysis. New York London: Academic Press 1984

    Google Scholar 

  5. Liao, M.: The existence of isometric stochastic flows for Riemannian Brownian motions. In: Pinsky, M., Wihstutz (eds.) Diffusion processes and the related problems in analysis, vol. II. Boston Basel Stuttgart: Birkhäuser 1992

    Google Scholar 

  6. Liao, M.: The Brownian motion and the canonical stochastic flow on a symmetric space. Trans. Am. Math. Soc. (to appear)

  7. Liao, M.: Stochastic flows on the boundaries of Lie groups. Stochastics Stochastic Rep.39, 213–237 (1992)

    Google Scholar 

  8. Malliavin, M.P., Malliavin, P.: Factorisations et lois limites de la diffusion horizontale au-dessus d'un espace Riemannien symmetrique. In: Théorie du Potentiel et Analyse Harmonique. (Lect. Notes Math., vol. 404, pp. 164–217) Berlin Heidelberg New York: Springer 1974

    Google Scholar 

  9. Norris, J.R., Rogers, L.C.G., Williams, D.: Brownian motion of ellipsoids. Trans. Am. Math. Soc.294, 757–765 (1986)

    Google Scholar 

  10. Orihara, A.: On random ellipsoid. J. Fac. Sci. Univ. Tyoko, Sect. IA Math.17, 73–85 (1970)

    Google Scholar 

  11. Prat, M.J.: Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative. C.R. Acad. Sci., Paris, Sér. A280, 1539–1524 (1975)

    Google Scholar 

  12. Taylor, J.C.: Brownian motion on a symmetric space of non-compact type: asymptotic behaviour in polar coordinates. Can. J. Math.43, 1065–1085 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics ACA, Aburn University, 36849, Auburn, AL, USA

    Ming Liao

Authors
  1. Ming Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by Hou Yin Dong Education Foundation of China On leave from Nankai University, Tianjin, China

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Liao, M. Stochastic flows on the boundaries of SL(n, R). Probab. Th. Rel. Fields 96, 261–281 (1993). https://doi.org/10.1007/BF01192136

Download citation

  • Received: 10 August 1992

  • Revised: 25 January 1993

  • Issue Date: June 1993

  • DOI: https://doi.org/10.1007/BF01192136

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (1991)

  • 58G32
  • 34D08
  • 58F10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature