The Positive Maximum Principle on Lie Groups

TL;DR

This paper generalizes the positive maximum principle to Lie groups, characterizing operators satisfying this principle as Levy type or pseudo-differential operators, depending on the group's properties.

Contribution

It extends Courr ext{e}ge's theorem to Lie groups, providing a global characterization of positive maximum principle operators.

Findings

Operators are Levy type with variable characteristics on Lie groups.

On compact groups, operators are pseudo-differential.

Constant characteristics imply generators of convolution semigroups.

Abstract

We extend a classical theorem of Courr\`{e}ge to Lie groups in a global setting, thus characterising all linear operators on the space of smooth functions of compact support that satisfy the positive maximum principle. We show that these are L\'{e}vy type operators (with variable characteristics), and pseudo--differential operators when the group is compact. If the characteristics are constant, then the operator is the generator of the contraction semigroup associated to a convolution semigroup of sub--probability measures.