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Potential Analysis

, Volume 12, Issue 3, pp 281–297 | Cite as

Longtime Behaviour of Stochastic Hamiltonian Systems: The Multidimensional Case

  • S. Albeverio
  • A. Klar
Article

Abstract

Hamiltonian systems perturbed by a white noise force are discussed in several dimensions. By using an appropriate scaling of the stochastic force a convergence theorem for the invariants of the deterministic motion is proved. This corresponds to convergence of the system to a stationary distribution. Especially motion in a central force field is considered; the energy and angular momentum processes are investigated.

Hamiltonian systems random perturbations stationary distribution invariants of motion nonlinear stochastic differential equations long time (asymptotic) behaviour multidimensional ergodic systems limit diffusions 

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References

  1. 1.
    Albeverio, S. and Klar, A.: ‘Longtime behaviour of nonlinear stochastic oscillators: The one dimensional Hamiltonian case’ J. Math. Phys. 35(8) (1994), 4005–4027.Google Scholar
  2. 2.
    Albeverio, S., Gong, Guanglu and Quian, Minping: On the Winding Numbers of Perturbed Linear Diffusions, BiBoS-Preprint, Bochum (1991).Google Scholar
  3. 3.
    Albeverio, S. and Hilbert, A.: Some Results on Newton Equation with an Additional Stochastic Force, in: N. Christopeit and K. Holmes (eds), Lecture Notes in Control and Information Sciences 126, Springer, 1989, pp. 1–13.Google Scholar
  4. 4.
    Albeverio, S., Hilbert, A. and Zehnder, E.: ‘Hamiltonian Systems with a stochastic force: Nonlinear versus linear and a Girsanov formula’ Stochastics and Stoch. Repts. 39 (1992) 159p.Google Scholar
  5. 5.
    Arnold, V. I.: Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978.Google Scholar
  6. 6.
    Hilbert, A.: Stochastic Perturbations of Hamiltonian Systems, Ph.D. Thesis, Bochum, 1990.Google Scholar
  7. 7.
    Klar, A.: Langzeitverhalten Nichtlinearer Stochastischer Oszillatoren, Diplomarbeit Ruhr-Universität Bochum, 1991.Google Scholar
  8. 8.
    Seesselberg, M., Breuer, H. P., Mais, H. Petruccione, F. and Honerkamp, J.: 'simulation of one-dimensional noisy Hamiltonian systems and their application to particle storage rings’ Z. Phys. C 62 (1994), 63–73.Google Scholar
  9. 9.
    Markus, L. and Weerasingle, A.: ‘Stochastic oscillators’ J. Diff. Equ. 21 (1988), 288–314.Google Scholar
  10. 10.
    Mc Kean, H. P.: Stochastic Integrals, Academic Press, New York, 1969.Google Scholar
  11. 11.
    Papanicolaou, G., Stroock, D. and Varadhan, S.: ‘Martingale approach to some limit theorems’ in: D. Ruelle (ed.), Statistical Mechanics and Dynamical Systems, Duke University Mathematics Series III, Durham, 1978.Google Scholar
  12. 12.
    Potter, J.: Some Statistical Properties of a Nonlinear Oscillator Driven by White Noise, Ph. D. Thesis, MIT (1962).Google Scholar
  13. 13.
    Reed, M. and Simon, B.: Methods of Modern Mathematical Physics II, Academic Press, New York (1972).Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • S. Albeverio
    • 1
    • 2
  • A. Klar
    • 3
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Inst. Ang. Math.Universität BonnBonnGermany SFB 237, Bochum-Essen-Düsseldorf; BiBoS Bielefeld; CERFIM Locarno, Switzerland
  3. 3.Fachbereich Mathematik und InformatikFU BerlinBerlinGermany

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