Renormalization analysis of catalytic Wright--Fisher diffusions
- Fleischmann, Klaus
- Swart, Jan M.
2010 Mathematics Subject Classification
- 82C28 82C22 60J60 60J80
- Renormalization, catalytic Wright-Fisher diffusion, rescaled iterates, catalyzing function, renormalization branching process, embedded particle system, extinction, unbounded growth, interacting diffusions, duality, coupling, Poisson-cluster, fixed point equation, universality class
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.
- Electron. J. Probab., 11 (2006) pp. 585-654.