On the Feller-Dynkin and the Martingale Property of One-Dimensional Diffusions

We show that a one-dimensional regular continuous Markov process $\text{X}$ with scale function $s$ is a Feller–Dynkin process precisely if the space transformed process $s(X)$ 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 multi-dimensional diffusions. Moreover, for It^o diffusions we discuss relations to Cauchy problems.