Renormalization analysis of catalytic Wright-Fisher diffusions
Jan M. Swart, UTIA
Klaus Fleischmann, WIAS Berlin
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.
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