December, 2024 Dirichlet form approach to one-dimensional Markov processes with discontinuous scales
Liping LI
Author Affiliations +
J. Math. Soc. Japan Advance Publication 1-37 (December, 2024). DOI: 10.2969/jmsj/92649264

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

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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

Information

Received: 10 December 2023; Revised: 17 April 2024; Published: December, 2024
First available in Project Euclid: 6 December 2024

Digital Object Identifier: 10.2969/jmsj/92649264

Subjects:
Primary: 31C25
Secondary: 60J35 , 60J45 , 60J46

Keywords: Diffusion processes , Dirichlet forms , quasidiffusions , Ray–Knight compactification , regular representations

Rights: Copyright ©2024 Mathematical Society of Japan

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