Central limit theorem for a system of Markovian particles with mean field interactions

  • Tokuzo Shiga
  • Hiroshi Tanaka
Article

Keywords

Stochastic Process Probability Theory Limit Theorem Mathematical Biology Central Limit 

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References

  1. 1.
    Dawson, D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31, 29–85 (1983)Google Scholar
  2. 2.
    Dellacherie, C., Meyer, P.A.: Probabilities et potentiel, Chapitres V a VIII. Herman: Paris 1980Google Scholar
  3. 3.
    Dynkin, E.B., Mandelbaum, A.: Symmetric statistics, Poisson point processes and multiple Wiener integrals. Ann. Statist. 11, 739–745 (1983)Google Scholar
  4. 4.
    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland/Kodansha 1981Google Scholar
  5. 5.
    Krylov, N.V.: On Ito's stochastic differential equations. Theory Probab. Appl. 14, 330–336 (1969)Google Scholar
  6. 6.
    Kunita, H., Watanabe, S.: On square integrable martingales. Nagoya Math. J. 30, 209–245 (1967)Google Scholar
  7. 7.
    Kusuoka, S., Tamura, Y.: Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo 31, 223–245 (1984)Google Scholar
  8. 8.
    Maruyama, G.: On the transition probability functions of the Markov processes. Nat. Sci. Rep. Ochanomizu Univ. 5, 10–20 (1954)Google Scholar
  9. 9.
    McKean, H.P.: Propagation of chaos for a class of non-linear parabolic equations. Lecture Series in Differential Equations, 7 Catholic Univ. 41–57 (1967)Google Scholar
  10. 10.
    McKean, H.P.: Fluctuations in the kinetic theory of gases. Commun. Pure Appl. Math., 28, 435–455 (1975)Google Scholar
  11. 11.
    Parthasarathy, K.R.: Probability measures on metric spaces. New York: Academic Press 1967Google Scholar
  12. 12.
    Simon, B.: Trace ideals and their applications. London Math. Soc. Lecture Note Series, 35. Cambridge: Univ. Press 1979Google Scholar
  13. 13.
    Sznitman, A.S.: Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56, 311–336 (1984)Google Scholar
  14. 14.
    Sznitman, A.S.: A fluctuation result for non-linear diffusions. [Preprint]Google Scholar
  15. 15.
    Tanaka, H.: Limit theorems for certain diffusion processes with interaction. Proceeding of the Taniguchi International Symposium on Stochastic Analysis (K. Itô ed.), pp. 469–488. Tokyo: Kinokuniya 1984Google Scholar
  16. 16.
    Tanaka, H., Hitsuda, M.: Central limit theorem for a simple diffusion model of interacting particles. Hiroshima Math. J. 11, 415–423 (1981)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Tokuzo Shiga
    • 1
  • Hiroshi Tanaka
    • 2
  1. 1.Department of Applied Physics, Faculty of ScienceTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of Mathematics, Faculty of Science and TechnologyKeio UniversityYokohamaJapan

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