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
Diffusive clustering in an infinite system of hierarchically interacting diffusions
Download PDF
Download PDF
  • Published: December 1994

Diffusive clustering in an infinite system of hierarchically interacting diffusions

  • Klaus Fleischmann1 &
  • Andreas Greven2 

Probability Theory and Related Fields volume 98, pages 517–566 (1994)Cite this article

  • 115 Accesses

  • 26 Citations

  • Metrics details

Summary

We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guaranties, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bramson, M., Cox, J. T., Griffeath, D.: Consolidation rates for two interacting systems in the plane. Probab. Theory Relat. Fields73, 613–625 (1986)

    Google Scholar 

  2. Chung, K. L.: Markov chains with stationary transition probabilities. 2nd edn Berlin, Heidelberg, New York: Springer 1967

    Google Scholar 

  3. Cox, J. T., Greven, A.: On the long term behavior of finite particle systems: a critical dimension example. In: Durrett, R., Kesten, H. (eds.) Random walks, Brownian motion and interacting particle systems. A festschrift in honor of Frank Spitzer. Progress in Probab.28, pp. 203–213, Boston: Birkhäuser 1991

    Google Scholar 

  4. Cox, J. T. Greven, A.: Ergodic theorems for inifinite systems of locally interacting diffusions. Ann. Probab. (to appear) (1994)

  5. Cox, J. T., Griffeath, D.: Occupation time limit theorem for the voter model. Ann. Probab.11, 876–893 (1983)

    Google Scholar 

  6. Cox, J. T., Griffeath, D.: Diffusive Clustering in the two dimensional voter model. Ann. Probab.14, 347–370 (1986)

    Google Scholar 

  7. Cox, J. T., Griffeath, D.: Mean field asymptotics for the planar stepping stone model. Proc. Lond. Math. Soc. III. Ser.61, 189–208 (1990)

    Google Scholar 

  8. Dawson, D. A., Greven, A.: Hierarchical models of interacting diffusions: Multiple time scale phenomena, phase transition and pattern of clusterformation. Probab. Theory Relat. Fields96, 435–473 (1993a)

    Google Scholar 

  9. Dawson, D. A., Greven, A.: Multiple time scale analysis of hierarchically interacting systems. Stochastic processes. A Festschrift in honor of Gopinath Kallianpur. Ed. S. Combanis et al. Berlin, Heidelberg, New York: Springer, pp. 41–50, 1993b

    Google Scholar 

  10. Erdös, P., Taylor, S. J.: Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hung.11, 137–162 (1960)

    Google Scholar 

  11. Ethier, S. N., Kurtz, T. G.: Markov processes: characterization and convergence. New York: Wiley 1986

    Google Scholar 

  12. Feller, W.: An introduction to probability theory and its applications, vol. II, 2nd edn. New York: Wiley 1971

    Google Scholar 

  13. Fleischmann, K.: Mixing properties of cluster-invariant distributions. Litov. Mat. Sb.18, 191–199 (1978)

    Google Scholar 

  14. Fleischmann, K., Greven, A.: Diffusive clustering in an infinite system of hierarchically interacting diffusions. Preprint No. 23, IAAS, Berlin 1992

    Google Scholar 

  15. Greven, A.: A phase transition for the coupled branching process, part 1: The ergodic theory in the range of finite second moments. Probab. Theory Relat. Fields87, 417–458 (1991)

    Google Scholar 

  16. Liggett, T. M.: Interacting particle systems. Berlin, Heidelberg, New York: Springer, 1985

    Google Scholar 

  17. Liggett, T. M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. verw. Gebiete56, 443–468 (1981)

    Google Scholar 

  18. Notohara, M., Shiga, T.: Convergence to genetically uniform state in stepping stone models of population genetics. J. Math. Biol.10, 281–294 (1980)

    Google Scholar 

  19. Ohta, T., Kimura, M.: A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population. Genet. Res. Camb.22, 201–204 (1973)

    Google Scholar 

  20. Sawyer, S., Felsenstein, J.: Isolation by distance in a hierarchically clustered population. J. Appl. Probab.20, 1–10 (1983)

    Google Scholar 

  21. Shiga, T.: An interacting system in population genetics. J. Math. Kyoto Univ.20, 213–242 (1980a)

    Google Scholar 

  22. Shiga, T.: An interacting system in population genetics, II. J. Math. Kyoto Univ.20, 723–739 (1980b)

    Google Scholar 

  23. Shiga, T., Shimizu, A.: Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ.20, 395–416 (1980)

    Google Scholar 

  24. Tavaré, S.: Line-of-descent and genealogical processes, and their applications in population genetics models. Theor. Popul. Biol.26, 119–164 (1984)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117, Berlin, Germany

    Klaus Fleischmann

  2. Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    Andreas Greven

Authors
  1. Klaus Fleischmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Greven
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fleischmann, K., Greven, A. Diffusive clustering in an infinite system of hierarchically interacting diffusions. Probab. Th. Rel. Fields 98, 517–566 (1994). https://doi.org/10.1007/BF01192837

Download citation

  • Received: 03 December 1992

  • Revised: 21 July 1993

  • Issue Date: December 1994

  • DOI: https://doi.org/10.1007/BF01192837

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

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Single Component
  • Qualitative Property
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