Generalisations of Lévy Operators to Manifolds and Symmetric Spaces

In this thesis, functional analytical methods are applied to the study of Lévy and Feller processes on manifolds. In the case of a compact Riemannian manifold, we prove that the Feller semigroup and generator of an isotropic Lévy process extend to Lᵖ, and are self-adjoint in the case p=2. When there is a non-trivial Brownian motion component to the process, we find that the generator has a discrete spectrum of non-positive eigenvalues, and that the semigroup is trace-class.

We also consider the case where the underlying manifold is a Riemannian symmetric space of noncompact type. Considering first the Lévy case, we use harmonic analysis to prove a sufficient condition for the associated convolution semigroup to possess an L² density, and calculate the spectrum of a self-adjoint Lévy generator. We then move on to consider Feller processes on a symmetric space of noncompact type. We develop a theory of pseudodifferential operators in this setting, prove that the semigroup and generator of a Feller process are both pseudodifferential operators in the sense we have defined, and calculate their symbols. Using the Hille--Yosida--Ray theorem, sufficient conditions are developed for a pseudodifferential operator to have a closed extension that generates a sub-Feller process. To demonstrate that these conditions are reasonable, we present a class of examples for which they are satisfied.