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...

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...

Large deviation principles are established for the Fleming-Viot processes with neutral mutation and selection, and the corresponding equilibrium measures as the sampling rate goes to 0. All results are first proved for the finite allele model, and then generalized, through the projective limit...

We obtain exact almost-sure estimates for the short-time propagation of the closed support of (2, d, [beta])-superprocesses. Upper estimates are derived by solving a certain singular non-linear evolution equation, whereas lower estimates are obtained by the use of the branching-particle-system...