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Parameter dependence of the Lyapunov exponent for linear stochastic systems. A survey

Part II: Linear Stochastic Systems. Stability Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1186)

Abstract

A survey is given on how formulas are obtained for the Lyapunov exponent of linear stochastic systems.

Keywords

  • Lyapunov Exponent
  • Invariant Measure
  • Elliptic Generator
  • Adjoint Problem
  • Continuous Time System

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Arnold, L.; Kliemann, W. and Oeljeklaus, E.: Lyapunov exponents of linear stochastic systems, in this volume

    Google Scholar 

  2. Arnold, L.; Papanicolaou, G. and Wihstutz, V.: Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and application; to appear in: SIAM J. Applied Mathematics. Preprint: Forschungsschwerpunkt Dynamische Systeme Universität Bremen, Report Nr. 134 (1985)

    Google Scholar 

  3. Auslender, E. I. and Mil'shtein, G. N.: Asymptotic expansion of the Liapunov index for linear stochastic systems with small noise, Prikl. Matem. Mekhan 46 (1982), 358–365; engl.: PMM U.S.S.R. 46 (1983), 277–283

    MathSciNet  Google Scholar 

  4. Avron, J.; Craig, W. and Simon, B.: Large coupling behavior of the Lyapunov exponent for tight binding one-dimensional random systems, J. Phys. A: Math. Gen. 16 (1983), L 209–211

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Cohen, J. E. and Newman, C. M.: The stability of large random matrices and their products, The Annals Prob. 12 (1984), 283–310

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Freidlin, M. I. and Wentzell, A. D.: Random perturbations of dynamical systems, engl.: Grundlehren vol. 260, Springer 1984

    Google Scholar 

  7. Furstenberg, H. and Kesten, H.: Products of random matrices, Annals of Math. Statist. 31 (1960), 457–469

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Furstenberg, H.: Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377–428

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Has'minskij, R. Z.: Stochastic stability of differential equations, engl.: Sijthoff & Noordhoff 1980 (russ. Moscow 1969)

    Google Scholar 

  10. Kato, T.: Perturbation theory for linear operators (second ed.), Springer 1980

    Google Scholar 

  11. Loparo, K. and Blankenship, G. L.: Almost sure instability of a class of linear stochastic systems with jump process coefficients, preprint 1983 and this volume, 1985

    Google Scholar 

  12. Nishioka, K.: On the stability of two-dimensional linear stochastic systems, Kodai Math. Sem. Rep. 27 (1976), 211–230

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Pardoux, E. and Wihstutz, V.: Lyapunov exponents of degenerated linear stochastic systems, preprint 1985

    Google Scholar 

  14. Pinsky, M. A.: Instability of the harmonic oscillator with small noise, preprint 1985

    Google Scholar 

  15. Schwartz, L.: Théorie des distributions, Hermann, Paris 1966

    MATH  Google Scholar 

  16. Wihstutz, V.: Analytic expansion of the Lyapunov exponent associated to the Schrödinger operator with random potential, Stochastic Analysis and Applications 3 (1985), 98–118

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1986 Springer-Verlag

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Wihstutz, V. (1986). Parameter dependence of the Lyapunov exponent for linear stochastic systems. A survey. In: Arnold, L., Wihstutz, V. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1186. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076840

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  • DOI: https://doi.org/10.1007/BFb0076840

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16458-6

  • Online ISBN: 978-3-540-39795-3

  • eBook Packages: Springer Book Archive