On generalization of quasidiffusions

Abstract: A quasidiffusion is by definition a time-changed Brownian motion on certain closed subset of $\mathbb{R}$. The aim of this paper is two-fold. On one hand, we will put forward a generation of quasidiffusion, called skip-free Hunt process, by way of a pathwise setup. As an analogue of regular diffusion on an interval, a skip-free Hunt process also admits a so-called scale function and a so-called speed measure. In addition, it is uniquely determined by scale function and speed measure. Particularly, a skip-free Hunt process is a quasidiffusion, if and only if it is on the natural scale. On the other hand, we will give an analytic characterization of skip-free Hunt processes by means of Dirichlet forms. The main result shows that a so-called strongly local-like property for associated Dirichlet forms is equivalent to the skip-free property of sample paths of skip-free Hunt processes. As a byproduct, the explicit expression of Dirichlet forms associated to skip-free Hunt processes will be formulated.