The model under consideration is a catalytic branching model constructed in Dawson and Fleischmann (1997), where the catalysts themselves undergo a spatial branching mechanism. The key result is a convergence theorem in dimension d = 3 towards a limit with full intensity (persistence), which, in...

For critical spatially homogeneous branching processes of finite intensity the following dichotomy is well-known: convergence to non-trivial steady states, or local extinction. In the latter case the underlying phenomenon is the growth of large clumps at spatially rare sites. For this situation...

A one-dimensional continuous measure-valued branching process is discussed, where branching occurs only at a single point catalyst described by the Dirac [delta]-function [delta]c. A (spatial) density field exists which is jointly continuous. At a fixed time t [greater-or-equal, slanted] 0, the...

Stochastic partial differential equations with polynomial coefficients have many applications in the study of spatially distributed populations in genetics, epidemiology, ecology, and chemical kinetics. The purpose of this paper is to describe some methods for investigating the qualitative...

We constructs a class of seperprocesses by taking the high density limit of a sequence of interacting-branching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching density is given by an arbitary bounded non-negative Borel...

A new formulation of non-local branching superprocesses is given from which we derive as special cases the rebirth, the multitype, the mass-structured, the multilevel and the age-reproduction-structured superprocesses and the superprocesscontrolled immigration process. This unied treatment...

Spatially homogeneous random evolutions arise in the study of the growth of a population in a spatially homogeneous random environment. The random evolution is obtained as the solution of a bilinear stochastic evolution equation. The main results are concerned with the asymptotic behavior of the...

We develop a general probabilistic approach that enables one to get sharp estimates for the almost-sure short-term behavior of hierarchically structured branching-diffusion processes. This approach involves the thorough investigation of the cluster structure and the derivation of some...