Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-31T11:45:38.301Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Attracting Orbit of a Non-Linear Transformation Arising From Renormalization Of Hierarchically Interacting Diffusions Part I: The Compact Case

Published online by Cambridge University Press:  20 November 2018

J. B. Baillon ,
PH. Clément ,
A. Greven  and
F. Den Hollander
J. B. Baillon
Affiliation:
Institut de Mathématiques et Informatique Université de Lyon I 43 Bd du 11 novembre 1918 F-69622 Villeurbanne Cedex France
PH. Clément
Affiliation:
Faculteit der Technische Wiskunde en Informatica Technische Universiteit DelftMekelweg 4 NL-2600 GA Delft The Netherlands
A. Greven
Affiliation:
Mathematisches Institut Universität Erlangen-NürnbergBismarckstrasse 1-½ D-91054 Erlangen Germany
F. Den Hollander
Affiliation:
Mathematisch Instituut Universiteit Nijmegen Toernooiveld1 NL-6525 ED Nijmegen The Netherlands
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.

Keywords

60K3547H99
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 1 , 01 February 1995 , pp. 3 - 27
Copyright
Copyright © Canadian Mathematical Society 1995

References

[B] Breiman, L., Probability, Addison-Wesley, Reading, 1968.Google Scholar
[CG] Cox, J. T. and Greven, A., On the long term behavior of finite particle systems: a critical dimension example. In: Random Walks, Brownian Motion and Interacting Particle Systems, A Festschrift in Honor of Frank Spitzer, (eds. Durrett, R. and Kesten, H.), Progr. Probab. 28, Birkhauser, Boston, 1991. 203–213.Google Scholar
[CGS] Cox, J.T., Greven, A. and Shiga, T., Finite and infinite systems of interacting diffusions, Probab. Theory Related Fields, to appear.Google Scholar
[DG1] Dawson, D.A. and Greven, A., Multiple time scale analysis of hierarchically interacting systems. In: Stochastic Processes, A Festschrift in Honour of Gopinath Kallianpur, (eds. Cambanis, S., Ghosh, J.K., Karandikar, R.L. and Sen, P.K.), Springer, New York, 1993. 41–50.Google Scholar
[DG2] Dawson, D.A. , Hierarchical models of interacting diffusions: multiple time scale phenomena, phase transition and pattern of cluster-formation, Probab. Theory Related Fields 96(1993), 435–473.Google Scholar
[DGV] Dawson, D.A., Greven, A. and Vaillancourt, J., Equilibria and quasi-equilibria for infinite collections of interacting Fleming-Viotprocesses, Trans. Amer. Math. Soc, to appear.Google Scholar
[D] Durrett, R., Lecture Notes on Particle Systems and Percolation, Wadsworth Brooks-Cole, Pacific Grove, 1988.Google Scholar
[FG] Fleischmann, K. and Greven, A., Diffusive clustering in an infinite system of hierarchically interacting diffusions, IAAS Berlin, 1992., Probab. Theory Related Fields 98(1994), 517–566.Google Scholar
[G] Georgii, H.-O., Gibbs Measures and Phase Transitions, Walter de Gruyter, Berlin, 1988.Google Scholar
[L] Liggett, T., Interacting Particle Systems, Springer, New York, 1985.Google Scholar
[OK] Ohta, T. and Kimura, M., A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population, Genet. Res. Camb. 22(1973), 201–204.Google Scholar
[S] Shiga, T., An interacting system in population genetics, J. Math. Kyoto Univ. 20(1980), I: 213-243, II: 723-733.Google Scholar
[SF] Sawyer, S. and Felsenstein, J., Isolation by distance in a hierarchically clustered population, J. Appl. Probab. 20(1983), 1–10.Google Scholar