On the Feller–Dynkin and the martingale property of one-dimensional diffusions

Abstract

We show that a one-dimensional regular continuous Markov process $\mathsf{X}$ with scale function $\mathfrak{s}$ is a Feller–Dynkin process precisely if the space transformed process $\mathfrak{s}(\mathsf{X})$ is a martingale when stopped at the boundaries of its state space. As a consequence, the Feller–Dynkin and the martingale property are equivalent for regular diffusions on natural scale with open state space. By means of a counterexample, we also show that this equivalence fails for multidimensional diffusions. Moreover, for Itô diffusions we discuss relations to Cauchy problems.