Interacting stochastic systems: longtime behavior and its renormalization analysis. (English) Zbl 0963.92030

Author’s abstract: We describe typical phenomena arising in the longtime behavior of interacting spatial stochastic systems and explain how they can be analyzed using the technique of renormalization by multiple space-time scales. We shall focus on models which arise in population genetics, in particular interacting Fisher-Wright diffusions. The main mathematical point is to give an approximate picture of the spatial stochastic system by passing to a large space-time scale view. This will lead to a simpler stochastic process called the interaction chain. The analysis of this object reduces mainly to the study of the orbit of iterations of a certain nonlinear map in function space. Properties of this orbit can be derived by finding fixed points or fixed shapes of the nonlinear map and by showing convergence properties of general orbits to the special ones generated by fixed points or fixed shapes.
An important point is that this analysis allows to explain the special role of certain specific stochastic models, which correspond to the fixed points and fixed shapes and which characterize a universality class of longtime behavior in a larger class of models. We continue with describing directions of further research. Most important is the problem of the extension of the renormalization technique to lattice models. We present a first step in this direction, the so-called finite systems scheme. Finally we outline the possible applications of the multi-scale analysis in mathematical biology, in particular evolution theory.


92D10 Genetics and epigenetics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
92D15 Problems related to evolution