On a Limit Theorem for Branching One-Dimensional Diffusion Processes

Abstract

Let M be a simply connected complete Riemannian manifold with constant sectional curvature, and consider a branching Brownian motion Y=(yt9Py) having M as the underlying state space ([5]), In the case of M = S, the ddimensional sphere, one can apply the results of Watanabe [16], [17] and Asmussen-Hering [1] to obtain a limit theorem on the number of particles in a domain for the process Y. If M= R, then, although M is not compact, the process F belongs to the class considered by Watanabe [17], and his argument works well. But, if M has constant negative sectional curvature — fc, then Y is not necessarily contained in the scheme of [17] and a new phenomenon appears. In this case, the Laplace-Beltrami operator is given by