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2006 Renormalization analysis of catalytic Wright-Fisher diffusions
Jan Swart, Klaus Fleischmann
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Electron. J. Probab. 11: 585-654 (2006). DOI: 10.1214/EJP.v11-341


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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Jan Swart. Klaus Fleischmann. "Renormalization analysis of catalytic Wright-Fisher diffusions." Electron. J. Probab. 11 585 - 654, 2006.


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

zbMATH: 1113.60082
MathSciNet: MR2242657
Digital Object Identifier: 10.1214/EJP.v11-341

Primary: 82C28
Secondary: 60J60 , 60J80 , 82C22

Keywords: catalytic Wright-Fisher diffusion , embedded particle system , extinction , Interacting diffusions , renormalization , unbounded growth , Universality

Vol.11 • 2006
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