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Invariant measures for nonlinear stochastic differential equations

Chapter 2: Nonlinear Random Dynamical Systems

Part of the Lecture Notes in Mathematics book series (LNM,volume 1486)

Keywords

  • Lyapunov Exponent
  • Invariant Measure
  • Stochastic Differential Equation
  • Ergodic Property
  • Strong Maximum Principle

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References

  1. L. Arnold and W. Kliemann (1983). Qualitative theory of stochastic systems. In Probabilistic Analysis and Related Topics (A. T. Bharucha-Reid, ed.) 3 1–79. Academic Press, New York.

    Google Scholar 

  2. L. Arnold, E. Oeljeklaus and E. Pardoux (1986). Almost sure and moment stability for linear Itô equations. In Lyapunov exponents (L. Arnold, V. Wihstutz, eds) Lect. Notes Math. 1186 129–159. Springer, Berlin Heidelberg New York.

    CrossRef  Google Scholar 

  3. R. Azencott (1974). Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France 102 193–240.

    MathSciNet  MATH  Google Scholar 

  4. P. H. Baxendale (1987). Moment stability and large deviations for linear stochastic differential equations. In Proc. Taniguchi Symposium on Probabilistic Methods in Mathematical Physics. Katata and Kyoto 1985. (N. Ikeda, ed.) 31–54. Kinokuniya, Tokyo.

    Google Scholar 

  5. P. H. Baxendale (1990). Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms. In Spatial Stochastic Processes: Festschrift for T. E. Harris (K. Alexander, J. Watkins, eds) Birkhauser, Boston Basel Stuttgart (in press).

    Google Scholar 

  6. P. H. Baxendale and D. W. Stroock (1988). Large deviations and stochastic flows of diffeomorphisms. Probab. Th. Rel. Fields 80 169–215.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. R. N. Bhattacharya (1978). Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab. 6 541–553.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. A. P. Carverhill (1985). Flows of stochastic dynamical systems: ergodic theory. Stochastics 14 273–317.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. A. Friedman (1973). Wandering out to infinity of diffusion processes. Trans. Am. Math. Soc. 184 185–203.

    MathSciNet  MATH  Google Scholar 

  10. R. Z. Khas'minskii (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Theory Probab. Appl. 5 179–196.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. R. Z. Khas'minskii (1967). Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems. Theory Probab. Appl. 12 144–147.

    CrossRef  MathSciNet  Google Scholar 

  12. R. Z. Khas'minskii (1980). Stochastic stability of differential equations. Sijthoff and Noordhoff, Alphen aan den Rijn.

    Google Scholar 

  13. W. Kliemann (1987). Recurrence and invariant measures for degenerate diffusions. Ann. Probab. 15 690–707.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. G. Maruyama and H. Tanaka (1959). Ergodic property of N-dimensional recurrent Markov processes. Mem. Fac. Sci. Kyushu Univ. A-13 157–172.

    MathSciNet  MATH  Google Scholar 

  15. V. I. Oseledec (1968). A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 197–231.

    MathSciNet  Google Scholar 

  16. D. Ruelle (1979). Ergodic theory of differential dynamical systems. Publ. Math. IHES 50 275–306.

    CrossRef  Google Scholar 

  17. D. W. Stroock (1986). On the rate at which a homogeneous diffusion approaches a limit, an application of the large deviation theory of certain stochastic integrals. Ann. Probab. 14 840–859.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. D. W. Stroock and S. R. S. Varadhan (1972). On the support of diffusion processes with applications to the strong maximum principle. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 3 333–359. Univ. California Press.

    MathSciNet  MATH  Google Scholar 

  19. D. W. Stroock and S. R. S. Varadhan (1979). Multidimensional diffusion processes. Springer, Berlin Heidelberg New York.

    MATH  Google Scholar 

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© 1991 Springer-Verlag

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Baxendale, P.H. (1991). Invariant measures for nonlinear stochastic differential equations. In: Arnold, L., Crauel, H., Eckmann, JP. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086663

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  • DOI: https://doi.org/10.1007/BFb0086663

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54662-7

  • Online ISBN: 978-3-540-46431-0

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