# The Positive Maximum Principle on Symmetric Spaces

## TL;DR

This paper extends the Courr extbackslash'ege theorem to symmetric spaces, introducing Gangolli operators that generalize Lévy process generators and characterizing their pseudo-differential nature on compact spaces.

## Contribution

It generalizes the positive maximum principle to symmetric spaces and introduces Gangolli operators, broadening the understanding of generators of Feller--Markov processes in this context.

## Key findings

- Gangolli operators satisfy the positive maximum principle.
- On compact symmetric spaces, Gangolli operators are pseudo-differential with scalar symbols.
- The framework applies to Lévy processes on symmetric spaces.

## Abstract

We investigate the Courr\`{e}ge theorem in the context of linear operators $A$ that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller--Markov processes. We also introduce Gangolli operators, which satisfy the positive maximum principle, and generalise the form associated with the generator of a L\'{e}vy process on a symmetric space. When the space is compact, we show that Gangolli operators are pseudo--differential operators having scalar symbols.