Ilie Grigorescu
University of Miami, Coral Gables, FL, USA

We present results on the asymptotic behavior of a system of interacting diffusions evolving in a bounded domain in ${\mathbb R}^{d}$ by tracing the empirical measure asymptotically, when the number of particles $N$, respectively time $t$, approach infinity. When $N\to \infty$, we obtain a hydrodynamic limit $\mu_{t}$ (Law of Large Numbers on the path space). This is the macroscopic profile and satisfies a semi-linear PDE with non-local boundary conditions. When $t\to\infty$, we obtain the quasi-stationary distributions (qsd) in explicit formula involving the resolvent of the Dirichlet kernel. The interaction is a hybrid between the Fleming-Viot branching diffusions and the Bak-Sneppen minimal fitness process. Like the original model, self-organizing criticality is present in the one-to-one mapping between the intensity of the branching mechanism and the family of qsd. Simple examples will illustrate the emergence of the concepts involved.