On Spatially Homogeneous Branching Processes in a Random Environment

✍ Scribed by Donald Dawson; Klaus Fleischmann

• Publisher: John Wiley and Sons
• Year: 1983
• Tongue: English
• Weight: 425 KB
• Volume: 113
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19831130124

✦ Synopsis

Spatially honiogeneous branching processes in Euclidean space Rd are usually defined in such a way that the evolution law Ex,: of a "particle" 6, a t position x Rd a t time t doesn't depend on x or t , of. MATTHES, KERSTAN and MECKE [8; Chapter 121 or DAWSON [2]. To model a homogeneous "random cnvironment" one can assume that E,,: is random in that {E,,t : x E Rd, t E R+} forms a random field which is invariant with respect to space and time translations. In the particular case of dimension d = 0 and discrete time such a model reduces to a GALTON-WATSON process in a random environment which was investigated by SMITE, WILKINSON, ATHREYA, KARLIN, KAPLAN, KEIDINO, NIELSEN, MOUNTFORD, TURNBAU (and probably others), cf.

given w the particle 6, generates a random number of offspring according to f, and the displacements of the particles are independent with common distribution I . Already in this simple case a number of interesting questions arise and it is the purpose of this paper to present some preliminary results.

1. The Deterministic Environment

Let G be the d-dimensional lattice group Zd, d 2 1, and 1 a fixed HAAR measure on G. Let N denote the set of all (non-negative) measures p on G which are integer-~ ~ ~~ l) Research supported by NSERC grant A 7750. 2, This paper was prepared during the visit of the second author to Carleton University in the fall of 1981.