Renormalisation of hierarchically interacting Cannings processes

Stochastic Analysis Seminar Series

In order to analyse universal patterns in the large space-time behaviour of interacting multi-type stochastic populations, a key approach has been to carry out a renormalisation analysis in the so-called hierarchical setting. This has provided considerable insight into the structure of interacting systems of finite-dimensional diffusions, such as Fisher-Wright or Feller diffusions, and their infinite-dimensional analogues, such as Fleming-Viot or Dawson-Watanabe superdiffusions.

This talk describes an extension of this work to a class of jump processes called Cannings processes. The key feature of the Cannings individual-based population model is that the offspring of a single individual can be a positive fraction of the total population. The interaction in the hierarchical version of the Cannings process comes from migration and reshuffling-resampling on all hierarchical space-time scales simultaneously.

A full renormalisation analysis is carried out in the so-called hierarchical mean-field limit. It turn out that there are four universality classes for the scaling behaviour on large space-time scales. These can be analysed with the help of compositions of certain M"obius-transformations.

We discuss the implications of the scaling, pointing out several new features. In particular, we obtain a full classification for when the system exhibits clustering (= develops spatially expanding mono-type regions), respectively, exhibits local coexistence (= allows for different types to live next to each other).

Location:
Speaker(s):

Frank den Hollander (Leiden University)

Date:
Monday, November 5, 2012 - 14:15
to 15:15