$$L^2$$ Properties of Lévy Generators on Compact Riemannian Manifolds

Abstract

AbstractWe consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an $${\mathbb {R}}^d$$ 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 $$L^p$$ L p , for $$1\le p<\infty $$ 1 ≤ p < ∞ . When $$p=2$$ 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.