The uniqueness of symmetrizing measure and linear diffusions

TL;DR

This paper investigates the uniqueness of symmetrizing measures in one-dimensional local Dirichlet spaces and provides a representation for such spaces along with conditions for subspace relations.

Contribution

It proves that irreducibility guarantees the uniqueness of symmetrizing measures and offers a new representation and criteria for Dirichlet space subspaces.

Findings

Irreducibility implies uniqueness of symmetrizing measure.

Provides a representation for 1D local, irreducible, regular Dirichlet spaces.

Establishes necessary and sufficient conditions for Dirichlet space subspaces.

Abstract

In this short article, we shall study one-dimensional local Dirichlet spaces. One result, which has its independent interest, is to prove that irreducibility implies the uniqueness of symmetrizing measure for right Markov processes. The other result is to give a representation for any 1-dim local, irreducible and regular Dirichlet space and a necessary and sufficient condition for a Dirichlet space to be regular subspace of another Dirichlet space.