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.