Recurrence and transience of branching diffusion processes on Riemannian manifolds (Q1872331)

Recurrence and transience of a branching diffusion process on a Riemannian manifold are related to some properties (including spectral properties) of a linear elliptic operator. It is shown that 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 later has a high enough intensity, then it may override the transience of the Brownian notion, 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 is found explicitly.