An infinite particle system with additive interactions

Abstract

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set as t → ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brownian Motions

“…The answer is yes if d>3 and no if d -\ or 2 (cf. [2], [4], [6] and for a related situation [3] [2] recently discovered the same result. The idea in this paper is to study what happens if one reverses the limit procedures just described (a similar but somewhat different limit problem was studied by Martm-Lof in [7], In fact a minor modification of our procedure can be used to prove his result).…”

“…((-R') W ) ; and In this section we again discuss processes of the sort introduced in section (2), only now we allow there to be infinitely many particles initially. The approach that we adopt mimicks, for our setting, the construction by Durrett [3].…”

A limit theorem for patch sizes in a selectively-neutral migration model

“…is not of standard type. Dawson and Ivanoff (1978) discuss analogs of (3.4) for d~3, and Durrett (1979) has similar results for a branching random walk model. See also Sawyer (1976b), Sawyer and Felsenstein (1980) and Sawyer and Fleischman (1979).…”