Levy Operators on Manifolds

The student will investigate the linear operators which arise as infinitesimal generators of semigroups associated to isotropic Levy processes on Riemannian manifolds in an $L^{2}$-framework. These are perturbations of the Laplace-Beltrami operator through non-local terms that arise as a superposition of jumps along geodesics. In particular she will seek
(i) Necessary and sufficient conditions for the operator to be self-adjoint and seek to get some information about its spectrum.
(ii) Necessary and sufficient conditions for the semigroup to be compact and trace-class.
(iii) Conditions for the underlying process to have a (regular) density, and a "trace formula" which relates the diagonal part of the density to the trace of the semigroup.
She will also study Feynman-Kac type perturbations of the generator through a potential term.
Finally, she should (a) Study similar properties for lift of the semigroup to the bundle of orthonormal frames, and find natural relationships between the two classes of semigroups.
(b) Study the induced semigroups and generators on spaces of $p$ forms.