Electronic Journal of Probability

Renormalization analysis of catalytic Wright-Fisher diffusions

Jan Swart and Klaus Fleischmann

Full-text: Open access

Abstract

Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions indexed by the hierarchical group, with a linear attractive interaction between the components. In this context, the mappings are called renormalization transformations. We consider such maps for catalytic Wright-Fisher diffusions. These are diffusions on the unit square where the first component (the catalyst) performs an autonomous Wright-Fisher diffusion, while the second component (the reactant) performs a Wright-Fisher diffusion with a rate depending on the first component through a catalyzing function. We determine the limit of rescaled iterates of renormalization transformations acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 24, 585-654.

Dates
Accepted: 3 August 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730559

Digital Object Identifier
doi:10.1214/EJP.v11-341

Mathematical Reviews number (MathSciNet)
MR2242657

Zentralblatt MATH identifier
1113.60082

Subjects
Primary: 82C28: Dynamic renormalization group methods [See also 81T17]
Secondary: 82C22: Interacting particle systems [See also 60K35] 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Renormalization catalytic Wright-Fisher diffusion embedded particle system extinction unbounded growth interacting diffusions universality

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Swart, Jan; Fleischmann, Klaus. Renormalization analysis of catalytic Wright-Fisher diffusions. Electron. J. Probab. 11 (2006), paper no. 24, 585--654. doi:10.1214/EJP.v11-341. https://projecteuclid.org/euclid.ejp/1464730559


Export citation

References

  • J.-B. Baillon, Ph. Clément, A. Greven, and F. den Hollander. On the attracting orbit of a non-linear transformation arising from renormalization of hierarchically interacting diffusions. I. The compact case. Canad. J. Math. 47(1) (1995) 3-27.
  • J.-B. Baillon, Ph. Clément, A. Greven, and F. den Hollander. On the attracting orbit of a non-linear transformation arising from renormalization of hierarchically interacting diffusions. II. The non-compact case. J. Funct. Anal. 146 (1997) 236-298.
  • J.T. Cox, D.A. Dawson, and A. Greven. Mutually catalytic super branching random walks: Large finite systems and renormalization analysis. Mem. Am. Math. Soc. 809 (2004).
  • D.A. Darling and P. Erdös. On the recurrence of a certain chain. Proc. Am. Math. Soc. 19(1) (1968) 336-338.
  • D.A. Dawson and A. Greven. Hierarchical models of interacting diffusions: Multiple time scale phenomena, phase transition and pattern of cluster-formation. Probab. Theory Related Fields 96(4) (1993) 435-473.
  • D.A. Dawson and A. Greven. Multiple time scale analysis of interacting diffusions. Probab. Theory Related Fields 95(4) (1993) 467-508.
  • D.A. Dawson and A. Greven. Multiple space-time scale analysis for interacting branching models. Electron. J. Probab., 1 (1996) no. 14, approx. 84 pp.
  • D.A. Dawson, A. Greven and J. Vaillancourt. Equilibria and quasi-equilibria for infinite collections of interacting Fleming-Viot processes. Trans. Amer. Math. Soc. 347(7) (1995) 2277-2360.
  • F. den Hollander and J.M. Swart. Renormalization of hierarchically interacting isotropic diffusions. J. Stat. Phys. 93 (1998) 243-291.
  • S.N. Ethier and T.G. Kurtz. Markov Processes; Characterization and Convergence. John Wiley & Sons, New York, 1986.
  • N. El Karoui and S. Roelly. Propriétés de martingales, explosion et représentation de Lévy- Khintchine d'une classe de processus de branchement à valeurs mesures. Stoch. Proc. Appl. 38(2) (1991) 239-266.
  • W.J. Ewens. Mathematical Population Genetics. I: Theoretical Introduction. 2nd ed. Interdisciplinary Mathematics 27. Springer, New York, 2004.
  • P.J. Fitzsimmons. Construction and regularity of measure-valued branching processes. Isr. J. Math. 64(3) (1988) 337-361.
  • K. Fleischmann and J.M. Swart. Extinction versus exponential growth in a supercritical super-Wright-Fischer diffusion. Stoch. Proc. Appl. 106(1) (2003) 141-165.
  • K. Fleischmann and J.M. Swart. Trimmed trees and embedded particle systems. Ann. Probab. 32(3a) (2004) 2179-2221.
  • A. Greven, A. Klenke, and A. Wakolbinger. Interacting Fisher-Wright diffusions in a catalytic medium. Probab. Theory Related Fields 120(1) (2001) 85-117.
  • M. Jiřina. Branching processes with measure-valued states. In Trans. Third Prague Conf. Information Theory, Statist. Decision Functions, Random Processes (Liblice, 1962), pages 333-357, Czech. Acad. Sci., Prague, 1964.
  • O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, 1976.
  • A. Klenke. Different clustering regimes in systems of hierarchically interacting diffusions. Ann. Probab. 24(2) (1996) 660-697.
  • A. Liemant. Kritische Verzweigungsprozesse mit allgemeinem Phasenraum. IV. Math. Nachr. 102 (1981) 235-254.
  • M. Loève. Probability Theory 3rd ed. Van Nostrand, Princeton, 1963.
  • M. Loève. Probability Theory II 4th ed. Graduate Texts in Mathematics 46. Springer, New York, 1978.
  • A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983.
  • L.C.G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Volume 2: Ito Calculus. Wiley, Chichester, 1987.
  • F. Schiller. Application of the Multiple Space-Time Scale Analysis on a System of R-valued, Hierarchically Interacting, Stochastic Differential Equations. Master thesis, Universtity Erlangen-Nürnberg, 1998.
  • S. Sawyer and J. Felsenstein. Isolation by distance in a hierarchically clustered population. J. Appl. Probab. 20 (1983) 1-10.
  • T. Shiga. An interacting system in population genetics. J. Math. Kyoto Univ. 20 (1980) 213-242.
  • J.M. Swart. Large Space-Time Scale Behavior of Linearly Interacting Diffusions. PhD thesis, Katholieke Universiteit Nijmegen, 1999.
  • J.M. Swart. Clustering of linearly interacting diffusions and universality of their long-time limit distribution. Prob. Theory Related Fields 118 (2000) 574-594.
  • T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 (1971) 155-167.