Dawson DA, Greven A (2003)
Publication Type: Journal article, Original article
Publication year: 2003
Publisher: Institute of Physics: Hybrid Open Access
Book Volume: 8
Pages Range: 1-93
Journal Issue: 4
URI: https://projecteuclid.org/euclid.ejp/1464037577
The paper focuses on spatial multitype branching systems with spatial components (colonies) indexed by
a countable group, for example Z d or the hierarchical group. As type space we allow continua and describe
populations in one colony as measures on the type space. The spatial components of the system interact
via migration. Instead of the classical independence assumption on the evolution of different families of the
branching population, we introduce interaction between the families through a state dependent branching rate of
individuals and in addition state dependent mean offspring of individuals. However for most results we consider
the critical case in this work. The systems considered arise as diffusion limits of critical multiple type branching
random walks on a countable group with interaction between individual families induced by a branching rate
and offspring mean for a single particle, which depends on the total population at the site at which the particle
in question is located.
The main purpose of this paper is to construct the measure valued diffusions in question, characterize them
via well-posed martingale problems and finally determine their longtime behavior, which includes some new
features. Furthermore we determine the dynamics of two functionals of the system, namely the process of
total masses at the sites and the relative weights of the different types in the colonies as system of interacting
diffusions respectively time-inhomogeneous Fleming-Viot processes. This requires a detailed analysis of path
properties of the total mass processes.
In addition to the above mentioned systems of interacting measure valued processes we construct the corre-
sponding historical processes via well-posed martingale problems. Historical processes include information on
the family structure, that is, the varying degrees of relationship between individuals.
Ergodic theorems are proved in the critical case for both the process and the historical process as well as the
corresponding total mass and relative weights functionals. The longtime behavior differs qualitatively in the
cases in which the symmetrized motion is recurrent respectively transient. We see local extinction in one case
and honest equilibria in the other.
This whole program requires the development of some new techniques, which should be of interest in a wider
context. Such tools are dual processes in randomly fluctuating medium with singularities and coupling for
systems with multi-dimensional components.
The results above are the basis for the analysis of the large space-time scale behavior of such branching
systems with interaction carried out in a forthcoming paper. In particular we study there the universality
properties of the longtime behavior and of the family (or genealogical) structure, when viewed on large space
and time scales.
APA:
Dawson, D.A., & Greven, A. (2003). State dependent multi-type spatial branching processes and their longtime behaviour. European Journal of Physics, 8(4), 1-93.
MLA:
Dawson, Donald Andrew, and Andreas Greven. "State dependent multi-type spatial branching processes and their longtime behaviour." European Journal of Physics 8.4 (2003): 1-93.
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