On extensions of local Dirichlet forms

TL;DR

This paper constructs extremal local Dirichlet forms extending a given form and characterizes the conditions under which these extensions are unique, with applications to set-theoretic distances and capacity estimates.

Contribution

It introduces minimal and maximal local Dirichlet forms extending a base form and provides criteria for their equality and uniqueness.

Findings

Constructed minimal and maximal extensions of Dirichlet forms.

Characterized extension forms via algebraic ideals and capacity estimates.

Established invariance of set-theoretic distances for certain local forms.

Abstract

Let $\ce$ be a Dirichlet form on $L_2(X\,;\mu)$ where $(X,\mu)$ is locally compact $\sigma$-compact measure space. Assume $\ce$ is inner regular, i.e.\ regular in restriction to functions of compact support, and local in the sense that $\ce(\varphi,\psi)=0$ for all $\varphi, \psi\in D(\ce)$ with $\varphi\,\psi=0$. We construct two Dirichlet forms $\ce_m$ and $\ce_M$ such that $\ce_m\leq \ce\leq \ce_M$. These forms are potentially the smallest and largest such Dirichlet forms. In particular $\ce_m\supseteq \ce_M$, $(\ce_M)_m=\ce_m$ and $(\ce_m)_M=\ce_M$. We analyze the family of local, inner regular, Dirichlet forms $\cf$ which extend $\ce$ and satisfy $\ce_m\leq \cf\leq \ce_M$. We prove that the latter bounds are valid if and only if $\cf_M=\ce_M$, or $\cf_m=\ce_m$, or $D(\ce_M)$ is an order ideal of $D(\cf)$. Alternatively the $\cf$ are characterized by $D(\ce_M)\cap L_\infty(X)$ being an algebraic ideal of $D(\cf)\cap L_\infty(X)$. As an application we show that if $\ce$ and $\cf$ are strongly local then the Ariyoshi--Hino set-theoretic distance is the same for each of the forms $\ce$, $\ce_M$ and $\cf$. If in addition $\ce_m$ is strongly local then it also defines the same distance. Finally we characterize the uniqueness condition $\ce_M=\ce_m$ by capacity estimates.