  We shed a new light on the $L^1$-Liouville property for positive,
superharmonic functions by providing many evidences that its validity relies on
geometric conditions localized on large enough portions of the space. We also
present examples in any dimension showing that the $L^1$-Liouville property is
strictly weaker than the stochastic completeness of the manifold. The main tool
in our investigations is represented by the potential theory of a manifold with
boundary subject to Dirichlet boundary conditions. The paper incorporates,
under a unifying viewpoint, some old and new aspects of the theory, with a
special emphasis on global maximum principles and on the role of the Dirichlet
Green's kernel.
