  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.
