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
Critical branching in a highly fluctuating random medium
Download PDF
Download PDF
  • Published: June 1991

Critical branching in a highly fluctuating random medium

  • Donald A. Dawson1 &
  • Klaus Fleischmann2 

Probability Theory and Related Fields volume 90, pages 241–274 (1991)Cite this article

  • 110 Accesses

  • 22 Citations

  • Metrics details

Summary

A model of one-dimensional critical branching (superprocess) is constructed in a random medium fluctuating both in time and space. The medium describes a moving system of point catalysts, and branching occurs only in the presence of these catalysts. Although the medium has an infinite overall density, the clumping features of the branching model can be exhibited by rescaling time, space, and mass by an exactly calculated scaling power which is stronger than in the constant medium case. The main technique used is the asymptotic analysis of a generalized diffusion-reaction equation in the space-time random medium, which (given the medium) prescribes the evolution of the Laplace transition functional of the Markov branching process.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Dawson, D.A.: Stochastic evolution equations and related measure processes. J. Multivariate Anal.5, 1–52 (1975)

    Google Scholar 

  2. Dawson, D.A., Fleischmann, K.: On spatially homogeneous branching processes in a random environment. Math. Nachr.113, 249–257 (1983)

    Google Scholar 

  3. Dawson, D.A., Fleischmann, K.: Critical dimension for a model of branching in a random medium. Z. Wahrscheinlichkeitstheor, Verw. Geb.70, 315–334 (1985)

    Google Scholar 

  4. Dawson, D.A., Fleischmann, K.: Strong clumping of critical space-time branching models in subcritical dimensions. Stochastic Processes Appl.30, 193–208 (1988)

    Google Scholar 

  5. Dawson, D.A., Fleischmann, K.: Diffusion and reaction caused by point-catalysts. SIAM J. Appl. Math. (in press)

  6. Dawson, D.A., Perkins, E.A.: Historical processes. Mem. Am. Math. Soc. (in press)

  7. Dawson, D.A., Fleischmann, K., Gorostiza, L.G.: Stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium. Ann. Probab.17, 1083–1117 (1989)

    Google Scholar 

  8. Dawson, D.A., Fleischmann, K., Roelly, S.: Absolute continuity of the measure states in a branching model with catalysts. In: Cinlar, E., Williams, R.J., Fitzsimmons, P.J. (eds.) Progress in Probab. 24, Seminar on Stochastic Processes 1990

  9. Dynkin, E.B.: Branching particle systems and superprocesses. Manuscript, Cornell University Ithaca (1990)

  10. Fitzsimmons, P.J.: Construction and regularity of measure-valued Markov branching processes. Isr. J. Math.64, 337–361 (1988)

    Google Scholar 

  11. Fitzsimmons, P.J.: Correction and addendum to: Construction and regularity of measurevalued Markov branching processes. Isr. J. Math. (in press)

  12. Fleischmann, K.: Critical behavior of some measure-valued processes. Math. Nachr.135, 131–141 (1988)

    Google Scholar 

  13. Fleischmann, K., Gärtner, J.: Occupation time processes at a critical point. Math. Nachr.125, 275–290 (1986)

    Google Scholar 

  14. Fleischmann, K., Greven, A.: Localization and selection in a mean field branching random walk in a random environment. Universität Heidelberg, SFB 123, Preprint 591 (1990)

  15. Fleischmann, K., Molchanov, S.A.: Exact asymptotics in a mean field model with random potential. Probab. Th. Rel. Fields86, 239–251 (1990)

    Google Scholar 

  16. Kallenberg, O.: Random measures. 3rd revised and enlarged ed. Berlin: Akademie 1983

    Google Scholar 

  17. Kesten, H., Spitzer, F.: A limit theorem related to a new class of self-similar processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.50, 5–25 (1979)

    Google Scholar 

  18. Kozlov, S.M.: The method of averaging and walks in inhomogeneous environments. Russ. Math. Surv.40, 73–145 (1985)

    Google Scholar 

  19. Lamperti, J.: The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.7, 271–288 (1967)

    Google Scholar 

  20. Perkins, E.A.: A space-time property of a class of measure-valued branching diffusions. Trans. Am. Math. Soc.305, 743–795 (1988)

    Google Scholar 

  21. Roelly-Coppoletta, S.: A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics17, 43–65 (1986)

    Google Scholar 

  22. Sinai, Ja.G.: Limit behavior of one-dimensional random walks in a random medium. Theory Probab. Appl.27, 256–268 (1982)

    Google Scholar 

  23. Watanabe, S.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ.8, 141–167 (1968)

    Google Scholar 

  24. Yosida, K.: Functional analysis. 5-th edn. Berlin: Springer 1978

    Google Scholar 

  25. Zähle, U.: Self-similar random measures, I. notion, carrying Hausdorff dimension and hyperbolic distribution. Probab. Th. Rel. Fields80, 79–100 (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Carleton University, K1S 5B6, Ottawa, Canada

    Donald A. Dawson

  2. Karl Weierstrass Institute of Mathematics, Mohrenstrasse 39, O-1086, Berlin, Germany

    Klaus Fleischmann

Authors
  1. Donald A. Dawson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Klaus Fleischmann
    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

Dawson, D.A., Fleischmann, K. Critical branching in a highly fluctuating random medium. Probab. Th. Rel. Fields 90, 241–274 (1991). https://doi.org/10.1007/BF01192164

Download citation

  • Received: 17 April 1989

  • Revised: 01 May 1991

  • Issue Date: June 1991

  • DOI: https://doi.org/10.1007/BF01192164

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
  • Asymptotic Analysis
  • Random Medium
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