Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Diffusion with interactions and collisions between coloured particles and the propagation of chaos
Download PDF
Download PDF
  • Published: June 1987

Diffusion with interactions and collisions between coloured particles and the propagation of chaos

  • Masao Nagasawa1 &
  • Hiroshi Tanaka2 

Probability Theory and Related Fields volume 74, pages 161–198 (1987)Cite this article

  • 198 Accesses

  • 27 Citations

  • Metrics details

Summary

One considers a system ofN particles on the real line which are of two different types (colours). Their dynamics is given by a stochastic differential equation with constant diffusion part; the drift felt by a particle of either type depends on the empirical measures of type 1 and 2 particles at every instant; further a reflection condition is imposed so that particles of different type are not allowed to cross each other. The article studies the Vlasov-McKean limit of the system asN→∞: propagation of chaos and an evolution equation for the limiting empirical measures is established, from where in particular an equation for the separating front between the two types follos.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Dunford, N., Schwartz, J.: Linear operators, vol. 1. New York: Interscience 1958

    Google Scholar 

  2. Harris, T.E.: Diffusion with “collisions” between particles. J. Appl. Probab.,2, 323–338 (1965)

    Google Scholar 

  3. Kusuoka, S., Tamura, Y.: Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo Sect. 1A Math.31, 223–245 (1984)

    Google Scholar 

  4. Maruyama, G.: On the transition probability functions of the Markov processes. Nat. Sci. Rep. Ochanomizu Univ.5, 10–20 (1954)

    Google Scholar 

  5. McKean, H.P.: A class of Markov processes associated with non-linear parabolic equations. Proc. Natl. Acad. Sci.56, 1907–1911 (1966)

    Google Scholar 

  6. McKean, H.P.: Propagation of chaos for a class of non-linear parabolic equations. Lecture Series in Differential Equations. Washington, D.C.: Catholic Univ. 41–57 (1967)

    Google Scholar 

  7. Nagasawa, M.: Segregation of a population in an environment, J. Math. Biol.9, 213–235 (1980)

    Google Scholar 

  8. Nagasawa, M.: An application of segregation model for septation ofEscherichia coli. J. Theor. Biol.90, 445–455 (1981)

    Google Scholar 

  9. Nagasawa, M., Yasue, K.: A statistical model of mesons. Publication de l'Institut Recherche Mathématique Avancée (CNRS)33, 1–48 (1982/83) Université Lois Pasteur (Strasbourg)

    Google Scholar 

  10. Nagasawa, M., Tanaka, H.: A diffusion process in a singular mean-drift-field. Z. Wahrscheinlichkeitstheor. Verw. Geb.68, 247–269 (1985)

    Google Scholar 

  11. Nagasawa, M., Tanaka, H.: Propagation of chaos for diffusing particles of two types with singular mean field interaction. Z. Wahrscheinlichkeitstheor. Verw. Geb.71, 69–83 (1986)

    Google Scholar 

  12. Saisho, Y., Tanaka, H.: Stochastic differential equations for mutually reflecting Brownian balls. To appear in Osaka J. Math.

  13. Shiga, T., Tanaka, H.: Central limit theorem for a system of Markovian particles with mean field interaction. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 439–445 (1985)

    Google Scholar 

  14. Sznitman, A.S.: Non-linear reflecting diffusion processes, and propagation of chaos, and fluctuations associated. J. Funct. Anal.56, 311–336 (1984)

    Google Scholar 

  15. Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J.9, 163–177 (1979)

    Google Scholar 

  16. Tanaka, H.: Limit theorems for certain diffusion processes with interaction. In: Itô, K. (ed.) Stochastic analysis, pp. 469–488. Kinokuniya Co. Ltd., Tokyo, North-Holland 1984

    Google Scholar 

  17. Veretennikov, A.Ju.: On strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb.111 (1980). English transl. in Math. USSR Sbornik39, 387–403 (1981)

    Google Scholar 

  18. Wilks, S.S.: Mathematical statistics. New York: Wiley 1962

    Google Scholar 

  19. Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Mat. Sb.93 (1974). English trans. in Math. USSR Sb.22, 129–149 (1974)

    Google Scholar 

  20. Oelschläger, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab.12, 458–479 (1984)

    Google Scholar 

  21. Oelschläger, K.: A law of large numbers for modelately interacting diffusion processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 279–322 (1985)

    Google Scholar 

  22. Nagasawa, M.: Macroscopic, intermediate, microscopic and mesons. To appear in Lect. Notes Physics. Berlin Heidelberg New York: Springer

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik der Universität Zürich, Rämistrasse 74, CH-8001, Zürich, Switzerland

    Masao Nagasawa

  2. Department of Mathematics, Faculty of Science and Technology, Keio University, 223, Hiyoshi, Yokohama, Japan

    Hiroshi Tanaka

Authors
  1. Masao Nagasawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroshi Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

To the Memory of the late Professor Gishiro Maruýama

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nagasawa, M., Tanaka, H. Diffusion with interactions and collisions between coloured particles and the propagation of chaos. Probab. Th. Rel. Fields 74, 161–198 (1987). https://doi.org/10.1007/BF00569988

Download citation

  • Received: 23 October 1985

  • Revised: 01 September 1986

  • Issue Date: June 1987

  • DOI: https://doi.org/10.1007/BF00569988

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Colour
  • Reflection
  • Differential Equation
  • Stochastic Process
  • Probability Theory
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature