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...

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...