Pacific Institute for the Mathematical Sciences - PIMS

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.