Title:
Renormalization analysis of hierarchically interacting two-type branching models
Abstract: 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 via 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 review the basic renormalization program
and some recent progress (joint with Dawson, Greven, den Hoallander and Swart)
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. Open problems will also be
discussed.