Different clustering regimes in systems of hierarchically interacting diffusions

Abstract

We study a system of interacting diffusions $$\begin{eqnarray}dx_\xi(t)&=&\sum_{\zeta\in\Xi}a(\xi,\zeta)(x_\zeta(t)-x_\xi(t))dt \\ && + \sqrt{g(x_\xi(t))} dW_\xi(t) \qquad (\xi\in\Xi),\end{eqnarray}$$ indexed by the hierarchical group $\Xi$, as a genealogical two genotype model [where $x _\xi(t)$ denotes the frequency of, say, type A] with hierarchically determined degrees of relationship between colonies. In the case of short interaction range it is known that the system clusters. That is, locally one genotype dies out. We focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel $a(\dot\quad,\dot\quad)$ via its recurrent potential kernel. One of these regimes will be further investigated by mean-field methods. For general interaction range we shall also relate the behavior of large finite systems, indexed by finite subsets $\Xi_n$ of, $\Xi$ to that of the infinite system. On the way we will establish relations between hitting times of random walks and their potentials.