L2 Properties of Lévy Generators on Compact Riemannian Manifolds

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an R d -valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on L p , for 1 ≤ p < ∞ . When p = 2 , we show that these semigroups are self-adjoint. If, in addition, the motion has a non-trivial Brownian part, we prove that the generator has a discrete spectrum of eigenvalues and that the semigroup is trace-class.