The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.

Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.

The purpose of this paper is to relate the recurrence and transience properties of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator on M (including spectral properties). There is a trade-off 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 intensitythenitmayoverride 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.

Communicated by Heinz Siedentop Abstract. The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. 2000 Mathematics Subject Classification: 35P05, 81Q10