# $L^2$ Properties of L\'{e}vy Generators on Compact Riemannian Manifolds

## TL;DR

This paper studies the mathematical properties of isotropic Lévy processes on compact Riemannian manifolds, focusing on their semigroup extensions, spectral characteristics, and trace-class behavior, especially when combined with Brownian motion.

## Contribution

It establishes the extension of associated semigroups to $L^p$ spaces, their self-adjointness at $p=2$, and spectral properties when a Brownian component is present.

## Key findings

- Semigroups extend to strongly continuous contraction semigroups on $L^p$.
- Semigroups are self-adjoint on $L^2$.
- Generator has discrete spectrum and is trace-class with Brownian component.

## Abstract

We consider isotropic L\'evy processes on a compact Riemannian manifold, obtained from an $\mathbb{R}^d$-valued L\'evy 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 $1\leq p<\infty$, and that they are self-adjoint when $p=2$. When 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.