The renormalization transformation for two-type branching models

D. A. Dawson; A. Greven; F. den Hollander; Rongfeng Sun; J. M. Swart

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 6, page 1038-1077
  • ISSN: 0246-0203

Abstract

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This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space–time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space–time scaling.

How to cite

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Dawson, D. A., et al. "The renormalization transformation for two-type branching models." Annales de l'I.H.P. Probabilités et statistiques 44.6 (2008): 1038-1077. <http://eudml.org/doc/78002>.

@article{Dawson2008,
abstract = {This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space–time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space–time scaling.},
author = {Dawson, D. A., Greven, A., den Hollander, F., Sun, Rongfeng, Swart, J. M.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {interacting diffusions; space–time renormalization; two-type populations; independent branching; catalytic branching; mutually catalytic branching; universality; space-time renormalization},
language = {eng},
number = {6},
pages = {1038-1077},
publisher = {Gauthier-Villars},
title = {The renormalization transformation for two-type branching models},
url = {http://eudml.org/doc/78002},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Dawson, D. A.
AU - Greven, A.
AU - den Hollander, F.
AU - Sun, Rongfeng
AU - Swart, J. M.
TI - The renormalization transformation for two-type branching models
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 6
SP - 1038
EP - 1077
AB - This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space–time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space–time scaling.
LA - eng
KW - interacting diffusions; space–time renormalization; two-type populations; independent branching; catalytic branching; mutually catalytic branching; universality; space-time renormalization
UR - http://eudml.org/doc/78002
ER -

References

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