Abstract:  Courrege in 1965 obtained an interesting formula for a general linear operator on an open subset of a Euclidean space that satisfies the positive maximum principle. The formula resembles the generator of a Levy process, but has a much more general form. Because the generator of a Markov process satisfies the positive maximum principle, this immediately leads to a useful formula for the generator of a general type of Markov process. I will describe this formula, its extension to manifolds, and the relations of its different parts to the Markov process.