We relate the recurrence and transience of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator onM (including spectral properties). There is a trade-off between the tendency of the transient Brownian motion to escape 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.
"Recurrence and transience of branching diffusion processes on Riemannian manifolds." Ann. Probab. 31 (1) 244 - 284, January 2003. https://doi.org/10.1214/aop/1046294311