The Positive Maximum Principle on Lie Groups and Symmetric Spaces

In this thesis we will use harmonic analysis to get new results in probability on Lie groups and symmetric spaces. We will establish necessary and sufficient conditions for the existence of a square integrable K-bi-invariant density of a K-bi-invariant measure. We will show that there is a topological isomorphism between K-bi-invariant smooth functions and a subspace of the Sugiura space of rapidly decreasing functions. Furthermore, we will extend Courrège’s classical results to Lie groups and symmetric spaces, this consists of characterizing all linear operators on the space of smooth functions with compact support, that satisfy the positive maximum principle, as Lévy- type operators. We will specify some conditions under which such operators map to the Banach space of continuous functions vanishing at infinity, this allows us to study Feller semigroups and their generator in this context. We will show that on compact Lie groups all linear operators satisfying the positive maximum principle can be represented as pseudo-differential operators and on compact symmetric spaces they have analogous representations called spherical pseudo-differential operators