  We study recurrence and transience for L\'{e}vy processes induced by
topological transformation groups. 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 L\'{e}vy
processes acting in irreducible Riemannian symmetric pairs of non-compact type
are transient. We show that we also have "harmonic transience", i.e. local
integrability of the inverse of the real part of the characteristic exponent
which is associated to the process by means of Gangolli's L\'{e}vy-Khinchine
formula.
