A law of large numbers for moderately interacting diffusion processes

  • Karl Oelschläger
Article

Summary

We consider two special models of interacting diffusion processes, and derive in the limit, as the number of different processes tends to infinity and the interaction is rescaled in a suitable (“moderate”) way, a law of large numbers for the empirical processes. As limit dynamics we obtain certain nonlinear diffusion equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Braun, W., Hepp, K.: The Vlasov Dynamics and its Fluctuations in the 1/N Limit of Interacting Classical Particles. Comm. Math. Phys.56, 101–113 (1977)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Calderoni, P., Pulvirenti, M.: Propagation ofchaos for Burgers Equation. Ann. Inst. H. Poincaré, Sec. A, Physique Théorique,29, 85–97 (1983)MathSciNetGoogle Scholar
  3. 3.
    Cole, J.D.: On a quasilinear parabolic equation occuring in aerodynamics. Quart. Appl. Math.9, 225–236 (1951)MATHMathSciNetGoogle Scholar
  4. 4.
    Dudley, R.M.: Convergence of Baire Measures. Studia Mathematica27, 251–268 (1966)MATHMathSciNetGoogle Scholar
  5. 5.
    Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes I. Berlin-Heidelberg-New York: Springer 1974Google Scholar
  6. 6.
    Gutkin, E., Kac, M.: Propagation of Chaos and the Burgers Equation. SIAM J. Appl. Math.43, 971–980 (1983)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. Amsterdam-Oxford-New York: North Holland 1981Google Scholar
  8. 8.
    Ladyzěnskaja, O.A., Solonnikov, V.A., Ural'ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs Vol. 23. American Mathematical Society, Providence, Rhode Island 1968Google Scholar
  9. 9.
    McKean, H.P.: Propagation of Chaos for a Class of Nonlinear Parabolic Equations. Lecture Series in Differential Equations 7, Catholic Univ. 41–57 (1967)Google Scholar
  10. 10.
    Oelschläger, K.: A Martingale Approach to the Law of Large Numbers for Weakly Interacting Stochastic Processes. Ann. of Probab.12, 458–479 (1984)MATHGoogle Scholar
  11. 11.
    Osada, H., Kotani, S.: Propagation of chaos for Burgers Equation. Preprint (1983)Google Scholar
  12. 12.
    Rost, H.: Hydrodynamik gekoppelter Diffusionen: Fluktuationen im Gleichgewicht. Dynamics and Processes, Bielefeld 1981. ed. Ph. Blanchard, L. Streit, Lecture Notes in Mathematics 1031. Berlin-Heidelberg-New York: Springer 1983Google Scholar
  13. 13.
    Spohn, H.: Kinetic Equations from Hamiltonian Dynamics: The Markovian Limit. Rev. Mod. Phys.52, 569–616 (1980)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sznitman, A.S.: An example of a Nonlinear Diffusion Process with Normal Reflecting Boundary Conditions and some related Limit Theorems. Preprint (1983)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Karl Oelschläger
    • 1
  1. 1.Sonderforschungsbereich 123Universität HeidelbergHeidelberg 1Federal Republic of Germany

Personalised recommendations