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We study recurrence and transience for L\'{e}vy processes induced bytopological transformation groups. In particular the transience-recurrencedichotomy in terms of potential measures is established and transience is shownto be equivalent to the potential measure having finite mass on compact setswhen the group acts transitively. It is known that all bi-invariant L\'{e}vyprocesses acting in irreducible Riemannian symmetric pairs of non-compact typeare transient. We show that we also have "harmonic transience", i.e. localintegrability of the inverse of the real part of the characteristic exponentwhich is associated to the process by means of Gangolli's L\'{e}vy-Khinchineformula.

Aspects of Recurrence and Transience for Levy Processes in Transformation Groups and Non-Compact Riemannian Symmetric Pairs