Stochastic properties of Riemannian manifolds and applications to PDE's

The aim of this note is to describe geometric conditions under
which a Riemannian manifold enjoys the Feller property and to show
how the validity of the Feller property in combination with
stochastic completeness provides a new viewpoint to study
qualitative properties of solutions of semilinear elliptic PDE's
defined outside a compact set.