Effective intervals and regular Dirichlet subspaces

Liping Li, Wenjie Sun and Jiangang Ying

Stochastic Processes and their Applications, 2020, vol. 130, issue 10, 6064-6093

Abstract: It is shown in Li and Ying (2019) that a regular and local Dirichlet form on an interval can be represented by so-called effective intervals with scale functions. This paper focuses on how to operate on effective intervals to obtain regular Dirichlet subspaces. The first result is a complete characterization for a Dirichlet form to be a regular subspace of such a Dirichlet form in terms of effective intervals. Then we give an explicit road map how to obtain all regular Dirichlet subspaces from a local and regular Dirichlet form on an interval, by a series of intuitive operations on the effective intervals in the representation above. Finally applying previous results, we shall prove that every regular and local Dirichlet form has a special standard core generated by a continuous and strictly increasing function.

Keywords: Dirichlet forms; Regular Dirichlet subspaces; One-dimensional symmetric diffusions; Scale functions (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spa.2020.05.003

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