Electronic Journal of Probability

Renormalization analysis of catalytic Wright-Fisher diffusions

Jan Swart and Klaus Fleischmann

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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.

Article information

Electron. J. Probab., Volume 11 (2006), paper no. 24, 585-654.

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

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Zentralblatt MATH identifier

Primary: 82C28: Dynamic renormalization group methods [See also 81T17]
Secondary: 82C22: Interacting particle systems [See also 60K35] 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Renormalization catalytic Wright-Fisher diffusion embedded particle system extinction unbounded growth interacting diffusions universality

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Swart, Jan; Fleischmann, Klaus. Renormalization analysis of catalytic Wright-Fisher diffusions. Electron. J. Probab. 11 (2006), paper no. 24, 585--654. doi:10.1214/EJP.v11-341. https://projecteuclid.org/euclid.ejp/1464730559

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