We study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.
