Regular subspaces of Dirichlet forms

TL;DR

This paper investigates the structure, existence, and characterization of regular subspaces of Dirichlet forms, revealing that their jumping and killing measures remain unchanged from the original form, and employs probabilistic methods for local forms.

Contribution

It provides a detailed analysis of the structure and conditions for the existence of regular subspaces of Dirichlet forms, including their measure properties and probabilistic characterizations.

Findings

Jumping and killing measures are preserved in regular subspaces.

Established conditions for the existence of regular subspaces.

Used probabilistic transformations to characterize local Dirichlet subspaces.

Abstract

The regular subspaces of a Dirichlet form are the regular Dirichlet forms that inherit the original form but possess smaller domains. The two problems we are concerned are: (1) the existence of regular subspaces of a fixed Dirichlet form, (2) the characterization of the regular subspaces if exists. In this paper, we will first research the structure of regular subspaces for a fixed Dirichlet form. The main results indicate that the jumping and killing measures of each regular subspace are just equal to that of the original Dirichlet form. By using the independent coupling of Dirichlet forms and some celebrated probabilistic transformations, we will study the existence and characterization of the regular subspaces of local Dirichlet forms.