Abstract
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.
Funding Statement
The author is a member of LMNS, Fudan University. He is also partially supported by NSFC (No. 11931004 and 12371144) and Alexander von Humboldt Foundation in Germany.
Citation
Liping LI. "Dirichlet form approach to one-dimensional Markov processes with discontinuous scales." J. Math. Soc. Japan Advance Publication 1 - 37, December, 2024. https://doi.org/10.2969/jmsj/92649264
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