Dirichlet form approach to one-dimensional Markov processes with discontinuous scales

In this article, we will investigate a generalization of the Dirichlet form associated with a one-dimensional diffusion process. In this generalization, the scale function, which determines the expression of the Dirichlet form, is only required to be non-decreasing. While this generalized form is almost a Dirichlet form, it does not satisfy regularity in general. Consequently, it cannot be directly associated with a process in probability theory. To tackle this issue, we adopt Fukushima's regular representation method, which enables to find a family of strong Markov processes that are homeomorphic to each other and related to the generalized form in a certain sense. Additionally, this correspondence reveals the connection between this generalized form and a quasidiffusion. Moreover, we interpret the probabilistic implications behind the regular representation through two intuitive transformations. These transformations offer us the opportunity to obtain another symmetric non-strong Markov process with continuous sample paths. The Dirichlet form of this non-strong Markov process is precisely the non-regular generalized form we previously analyzed. Furthermore, the strong Markov process obtained from the regular representation is its Ray–Knight compactification.