Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brownian Motions

Abstract

The origins of this paper lie in a question posed to us by Frank Spitzer who, in fact, ended up solving most of his problem on his own. His problem is the following. Consider an infinite system of independent (^-dimensional branching Brownian motions which at their branching times disappear or double with equal probabilities, and assume that the initial distribution of the system is a Poisson point process. Denote by 5?,(-O, *^0 and r^ oo. The answer is yes if d>3 and no if d — or 2 (cf. [2], [4], [6] and for a related situation [3]). The question in which Spitzer was interested is what happens if (when d>3) one appropriately rescales the limiting random variable 5700. To be precise, given tf^>0, define for bounded