Renormalization analysis of catalytic Wright-Fisher diffusions

Details Renormalization analysis of catalytic Wright-Fisher diffusions Renormalization analysis of catalytic Wright-Fisher diffusions

2005-06-15

arxiv.org/abs/math/0506311v1

Electronic J. Probab. 11 (2006) paper no. 25, 585-654

Mathematics

Probability

65 pages, 3 figures

Scientific paper

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.

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