Markov and Feller Semigroups

Abstract

Markov and Feller semigroups are introduced, together with the corresponding stochastic processes. As all generators of Feller semigroups satisfy the positive maximum principle, we focus on that property and discuss the associated Hille–Yosida–Ray theorem. The main result of the chapter is proof of the Courrege theorem, which gives a Levy–Khinchine representation (but with variable coefficients) for all linear operators satisfying the positive maximum principle. We conclude with a brief discussion of the martingale problem and sub-Feller semigroups.