Conditional limit distributions of critical branching Brownian motions (Q1174212)

The long-time behavior of critical branching Brownian motions in \(\mathbb{R}^ d\) is studied under a variety of hypotheses. The main results are concerned with the limiting behavior conditioned on non-extinction in a bounded subset. Among the cases studied are those in which the motion process is a Bessel process, the critical offspring distribution belongs to the normal domain of attraction of a stable distribution with exponent \(p\), \(1<p\leq 2\), and the initial distribution is a single particle, finite intensity random field or an infinite intensity random field. The methods of the paper are analytical and are based on a scaling lemma and a comparison lemma for solutions to a nonlinear partial differential equation (the cumulant equation). The probabilistic significance of these results is related to their strong dimension dependence and in this sense extend previous results of \textit{D. A. Dawson}, \textit{K. Fleischmann}, \textit{R. D. Foley} and \textit{L. A. Peletier} [Stochastic Anal. Appl. 4, 117-129 (1986; Zbl 0588.60087)] and \textit{K. Fleischmann} [Math. Nachr. 82, 277- 296 (1978; Zbl 0316.60054); ibid. 84, 87-91 (1978; Zbl 0319.60054); and Stochastic differential systems, Proc. IFIP-WG 7/1 Work. Conf. Eisenach/GDR 1986, Lect. Notes Control Inf. Sci. 96, 22-26 (1987; Zbl 0641.60088)].