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
Perfect cocycles through stochastic differential equations
Download PDF
Download PDF
  • Published: March 1995

Perfect cocycles through stochastic differential equations

  • Ludwig Arnold1 &
  • Michael Scheutzow2 

Probability Theory and Related Fields volume 101, pages 65–88 (1995)Cite this article

  • 453 Accesses

  • 60 Citations

  • Metrics details

Summary

We prove that if ϕ is a random dynamical system (cocycle) for whicht→ϕ(t, ω)x is a semimartingale, then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment (helix), and conversely. This relation is succinctly expressed as “semimartingale cocycle=exp(semimartingale helix)”. To implement it we lift stochastic calculus from the traditional one-sided time ℝ to two-sided timeT=ℝ and make this consistent with ergodic theory. We also prove a general theorem on the perfection of a crude cocycle, thus solving a problem which was open for more than ten years.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Arnold, L.: Random dynamical systems (in preparation)

  2. Baxendale, P.: Brownian motion in the diffeomorphism group I. Compos. Math.53, 19–50 (1984)

    Google Scholar 

  3. Bismut, J.-M.: Méchanique aléatoire. (Lect. Notes Math. vol. 866) Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  4. Carmona, R., Nualart, D.: Nonlinear stochastic integrators, equations and flows. New York: Gordon and Breach 1990

    Google Scholar 

  5. Dudley, R.M.: Real analysis and probability. Wadsworth and Brooks/Cole, California: Pacific Grove 1989

    Google Scholar 

  6. Getoor, R.K.: Excessive measures, Boston: Birkhäuser 1990

    Google Scholar 

  7. Jacod, J.: Calcul stochastique et problèmes de martingales. (Lect. Notes Math., vol. 714), Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  8. Kolmogorov, A.N.: The Wiener helix, and other interesting curves in Hilbert space. Dokl. Akad. Nauk26, 115–118 (1940) (in Russian)

    Google Scholar 

  9. Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge: Cambridge University Press 1990

    Google Scholar 

  10. Le Jan, Y., Watanabe, S.: Stochastic flows of diffeomorphisms. In Itô, K. (ed.) Stochastic Analysis (Proceedings of the Taniguchi Symposium 1982). Amsterdam: North Holland 1984 pp. 307–332

    Google Scholar 

  11. Protter, P.: Semimartingales and measure preserving flows. Ann. Inst. Henri Poincaré (Probabilités et Statistiques)22, 127–147 (1986)

    Google Scholar 

  12. Protter, P.: Stochastic integration and differential equations. Berlin Heidelberg New York: Springer 1990

    Google Scholar 

  13. de Samlazaro, J., Meyer, P.A.: Questions des théorie des flots. Sémin. de Probab. IX, 1–153. (Lect. Notes. Math., vol. 465) Berlin Heidelberg New York: Springer 1975

    Google Scholar 

  14. Sharpe, M.: General theory of Markov processes. New York: Academic Press 1988

    Google Scholar 

  15. Walsh, J.B.: The perfection of multiplicative functionals. Sémin. de Probab. VI, 233–242. (Lect. Notes Math., vol. 258) Berlin Heidelberg New York: Springer 1972

    Google Scholar 

  16. Zimmer, R.J.: Ergodic theory and semisimple groups. Boston: Birkhäuser 1984

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Dynamische Systeme, Universität, Postfach 33 04 40, D028334, Bremen, Germany

    Ludwig Arnold

  2. Fachbereich Mathematik, Technische Universität, D-10623, Berlin, Germany

    Michael Scheutzow

Authors
  1. Ludwig Arnold
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Scheutzow
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This article was processed by the author using the latex style filepljour Im from Springer-Verlag.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arnold, L., Scheutzow, M. Perfect cocycles through stochastic differential equations. Probab. Th. Rel. Fields 101, 65–88 (1995). https://doi.org/10.1007/BF01192196

Download citation

  • Received: 04 October 1993

  • Revised: 28 June 1994

  • Issue Date: March 1995

  • DOI: https://doi.org/10.1007/BF01192196

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 Subjects Classification

  • 60H10
  • 93E03
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