Recurrence and transience of branching di usion processes on Riemannian manifolds

The purpose of this paper is to relate the recurrence and transience properties of a branching di usion process on a Riemannian manifold M to some properties of a linear elliptic operator on M (including spectral properties). There is a trade-o between the tendency of the Brownian motion to escape (if it is transient) and the birth process of the new particles. If the latter has a high enough intensity then it may override the transience of the Brownian motion, leading to the recurrence of the branching process, and vice versa. In the case of a spherically symmetric manifold, the critical intensity of the population growth can be found explicitly.