Renormalization analysis of catalytic Wright-Fisher diffusions. (English) Zbl 1113.60082

Fleischmann, Klaus; Swart, Jan M.

Electron. J. Probab. 11, Paper No. 24, 585-654 (2006).

Summary: 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.

MSC:

60J60	Diffusion processes
82C28	Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
82C70	Transport processes in time-dependent statistical mechanics