Communications in Mathematical Physics

, Volume 60, Issue 2, pp 153–170

The Onsager-Machlup function as Lagrangian for the most probable path of a diffusion process

  • Detlef Dürr
  • Alexander Bach
Article

Abstract

By application of the Girsanov formula for measures induced by diffusion processes with constant diffusion coefficients it is possible to define the Onsager-Machlup function as the Lagrangian for the most probable tube around a differentiable function. The absolute continuity of a measure induced by a process with process depending diffusion w.r.t. a quasi translation invariant measure is investigated. The orthogonality of these measures w.r.t. quasi translation invariant measures is shown. It is concluded that the Onsager-Machlup function cannot be defined as a Lagrangian for processes with process depending diffusion coefficients.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Graham, R.: Fluctuations, instabilities, and phase transitions. New York and London: Plenum Press 1975Google Scholar
  2. 2.
    Horsthemke, W., Bach, A.: Z. Physik B22, 189 (1975)Google Scholar
  3. 3.
    Haken, H.: Z. Physik B24, 321 (1976)Google Scholar
  4. 4.
    Enz, C.P.: Lecture notes in physics, Vol. 54, p. 54. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  5. 5.
    Graham, R.: Z. Physik B26, 281 (1977)Google Scholar
  6. 6.
    Bach, A., Dürr, D., Stawicki, B.: Z. Physik B26, 191 (1977)Google Scholar
  7. 7.
    Onsager, L., Machlup, S.: Phys. Rev.91, 1505 (1953);91, 1512 (1953)Google Scholar
  8. 8.
    Tisza, L., Manning, I.: Phys. Rev.105, 1695 (1957)Google Scholar
  9. 9.
    Dekker, H.: Physica85A, 363 (1976)Google Scholar
  10. 10.
    Kitahara, K., Metiu, H.: J. Stat. Phys.15, 141 (1976)Google Scholar
  11. 11.
    Stratonovich, R.L.: Sel. Trans. Math. Stat. Prob.10, 273 (1971)Google Scholar
  12. 12.
    Klauder, J.R.: Electromagnetic interactions and field theory. Wien-New York: Springer 1975Google Scholar
  13. 13.
    Streit, L.: Quantum electrodynamics. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  14. 14.
    Nelson, E.: Dynamical theory of Brownian motion. Princeton, NJ: Princeton University Press 1967Google Scholar
  15. 15.
    Gihman, I.I., Skorohod, A.V.: Stochastic differential equations. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  16. 16.
    Kuo, H.H.: Gaussian measures in Banach spaces. In: Lecture notes in mathematics, Vol. 463. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  17. 17.
    Yeh, J.: Stochastic processes and the Wiener integral. New York: Dekker 1973Google Scholar
  18. 18.
    Lipcer, R.S., Sirjaev, A.N.: Math. USSR Izv.6, 839 (1972)Google Scholar
  19. 19.
    Cameron, R.H., Martin, W.T.: Trans. Am. Math. Soc.66, 253 (1949)Google Scholar
  20. 20.
    Friedman, A.: Stochastic differential equations and applications, Vol. 1. New York-San Francisco-London: Academic Press 1975Google Scholar
  21. 21.
    Wong, E.: Stochastic processes in information and dynamical systems. New York: McGraw-Hill 1971Google Scholar
  22. 22.
    Gihman, I.I., Skorohod, A.V.: Theory of stochastic processes. I. Berlin-Heidelberg-New York: Springer 1974Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Detlef Dürr
    • 1
  • Alexander Bach
    • 1
  1. 1.Institut für Theoretische Physik I der UniversitätMünsterFederal Republic of Germany

Personalised recommendations