L^2 Properties of L\xe9vy Generators on Compact Riemannian Manifolds

Abstract

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an {mathbb {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 1le p<infty . 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.