The renormalization transformation for two-type branching models

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1038-1077.

Cet article étudie des systèmes dénombrables de diffusions en interaction hiérarchiques et linéaires vivant dans le quadrant positif. De tels systèmes apparaissent dans la dynamique d'individus de deux types qui migrent tout en interagissant dans des colonies. Le comportement à grande échelle et temps long peut être étudié en utilisant le programme de renormalisation. Ce programme, qui a permis de résoudre d'autres cas (principalement uni-dimensionnels) est basé sur la construction et l'analyse d'une transformation de renormalisation non linéaire, agissant sur la fonction de diffusion des composants du système et connectant l'évolution de blocs moyennés sur le temps à différentes échelles. Nous identifions une classe générale de fonctions de diffusion dans le quadrant positif pour lequel la transformation de renormalisation est bien définie et qui, sous une conjecture de comportement aux bords, peut-être itérée. À l'intérieur de certaines sous-classes, nous identifiens les points fixes de la transformation et étudions leurs domaines d'attraction. Ces domaines d'attraction constitutent les classes d'universalité du système après changement d'échelle dans le temps et l'espace.

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.

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