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**Multiple space-time scale analysis for interacting branching models.**
*(English)*
Zbl 0890.60093

Summary: We study a class of systems of countably many linearly interacting diffusions whose components take values in \([0, \infty)\) and which in particular includes the case of interacting (via migration) systems of Feller’s continuous state branching diffusions. The components are labelled by a hierarchical group. The longterm behaviour of this system is analysed by considering space-time renormalised systems in a combination of slow and fast time scales and in the limit as an interaction parameter goes to infinity. This leads to a new perspective on the large scale behaviour (in space and time) of critical branching systems in both the persistent and non-persistent cases and including that of the associated historical process. Furthermore we obtain an example for a rigorous renormalization analysis.

The qualitative behaviour of the system is characterised by the so-called interaction chain, a discrete time Markov chain on \([0,\infty)\) which we construct. The transition mechanism of this chain is given in terms of the orbit of a certain nonlinear integral operator. Universality classes of the longterm behaviour of these interacting systems correspond to the structure of the entrance laws of the interaction chain which in turn correspond to domains of attraction of special orbits of the nonlinear operator. There are two possible regimes depending on the interaction strength. We therefore continue in two steps with a finer analysis of the longtime behaviour. The first step focuses on the analysis of the growth of regions of extinction and the complementary regions of growth in the case of weak interaction and as time tends to infinity. Here we exhibit a rich structure for the spatial shape of the regions of growth which depend on the finer structure of the interaction but are universal in a large class of diffusion coefficients. This sheds new light on branching processes on the lattice. In a second step we study the family structure of branching systems in equilibrium in the case of strong interaction and construct the historical process associated with the interaction chain explicitly. In particular we obtain results on the number of different families per unit volume (as the volume tends to infinity). In addition we relate branching systems and their family structure (i.e. historical process) to the genealogical structure arising in Fleming-Viot systems. This allows us to draw conclusions on the large scale spatial distribution of different families in the limit of large times for both systems.

The qualitative behaviour of the system is characterised by the so-called interaction chain, a discrete time Markov chain on \([0,\infty)\) which we construct. The transition mechanism of this chain is given in terms of the orbit of a certain nonlinear integral operator. Universality classes of the longterm behaviour of these interacting systems correspond to the structure of the entrance laws of the interaction chain which in turn correspond to domains of attraction of special orbits of the nonlinear operator. There are two possible regimes depending on the interaction strength. We therefore continue in two steps with a finer analysis of the longtime behaviour. The first step focuses on the analysis of the growth of regions of extinction and the complementary regions of growth in the case of weak interaction and as time tends to infinity. Here we exhibit a rich structure for the spatial shape of the regions of growth which depend on the finer structure of the interaction but are universal in a large class of diffusion coefficients. This sheds new light on branching processes on the lattice. In a second step we study the family structure of branching systems in equilibrium in the case of strong interaction and construct the historical process associated with the interaction chain explicitly. In particular we obtain results on the number of different families per unit volume (as the volume tends to infinity). In addition we relate branching systems and their family structure (i.e. historical process) to the genealogical structure arising in Fleming-Viot systems. This allows us to draw conclusions on the large scale spatial distribution of different families in the limit of large times for both systems.

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |