## State dependent multitype spatial branching processes and their longtime behavior.(English)Zbl 1076.60084

The authors study multitype branching systems with spatial components (colonies) indexed by a countable group, for example $$Z^d$$ or the hierarchical group. They describe populations in one colony as measures on the type space. The spatial components of the system interact via migration. They introduce interaction between the families through a state dependent branching rate and mean offspring of individuals. The systems considered arise as diffusion limits of some critical multiple-type branching random walks on a countable group. The main purpose of this paper is to construct the measure-valued diffusions via well-posed martingale problems and to study their longtime behaviors. In addition, the authors determine the process of total masses at the sites and the relative weights of the different types in the colonies as a system of interacting diffusions and time-inhomogeneous Fleming-Viot processes, respectively. This requires a detailed analysis of the path properties of the total mass processes. They also construct the corresponding historical processes via well-posed martingale problems, which contain information on the varying degrees of relationship between the individuals. Ergodic theorems are proved in the critical case for both the measure-valued diffusion and the historical process. The longtime behavior differs in the two cases. One sees local extinction when the symmetrized spatial motion is recurrent and honest equilibria when the symmetrized motion is transient. The investigation requires new techniques such as dual processes in randomly fluctuating medium with singularities and coupling for systems with multi-dimensional components, which should be of interest in a wider context. The results obtained here are the basis for the analysis of large space-time scale behavior of the branching systems with interaction, which will be carried out in a forthcoming paper.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G57 Random measures 60G60 Random fields 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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