Renormalization analysis of hierarchically interacting two-type branching models

Linearly interacting diffusions model the evolution of colonies of populations. When the interaction kernel is of an appropriate form and the diffusions are indexed by the hierarchical group, the large scale space-time behavior of the system exhibits universality which can be fully characterized using a renormalization analysis first developed by Dawson and Greven. Such renormalization analysis has previously been successfully carried out for interacting diffusions which are either one-dimensional or live on a compact state space. In most of these cases, there is a single type of large scale space-time behavior and the system exhibits so-called full universality. In this talk, we give an overview of the renormalization program and then discuss recent progress in the analysis of hierarchically interacting two-type branching diffusions, where each diffusion lives on [0,infty)2 and the structure of the system's large scale space-time behavior is much richer. This is work in progress joint with
D. Dawon, A. Greven, F. den Hollander and J. M. Swart.