PUB - Publikationen an der Universität Bielefeld

It is shown that, for every noncompact parabolic Riemannian manifold and every nonpolar compact in , there exists a positive harmonic function on which tends to at infinity. (This is trivial for , easy for , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space , where constants are the only positive superharmonic functions and, for every nonpolar compact set , there is a symmetric (positive) Green function for . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines , and the line segments ).