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

Abstract

We show that a one-dimensional regular continuous strong Markov process \(X\) 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. Furthermore, for Ito diffusions we discuss relations to existence and uniqueness properties of Cauchy problems, and we identify the infinitesimal generator.