Effective intervals and regular Dirichlet subspaces

TL;DR

This paper characterizes how to derive all regular Dirichlet subspaces from effective intervals and scale functions, providing explicit operations and demonstrating the existence of a special standard core for such forms.

Contribution

It offers a complete characterization and explicit procedures for obtaining all regular Dirichlet subspaces via effective intervals and scale functions.

Findings

Complete characterization of regular Dirichlet subspaces

Explicit operations to generate subspaces from effective intervals

Existence of a special standard core for all regular local Dirichlet forms

Abstract

It is shown in [10] 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.