The orthogonal complements of $H^1(\mathbb{R})$ in its regular Dirichlet extensions

TL;DR

This paper characterizes the orthogonal complements of the Sobolev space $H^1(R)$ within regular Dirichlet extensions of one-dimensional Brownian motion, providing explicit formulas and their regular representations.

Contribution

It explicitly describes the orthogonal complements of $H^1(R)$ in the Dirichlet extension space and characterizes their structure via two associated spaces and the darning method.

Findings

Explicit formulas for functions in the orthogonal complement.

Description of the orthogonal complement via two specific spaces.

Regular representations of these spaces using the darning method.

Abstract

Consider the regular Dirichlet extension $(\mathcal{E},\mathcal{F})$ for one-dimensional Brownian motion, that $H^1(\mathbb{R})$ is a subspace of $\mathcal{F}$ and $\mathcal{E}(f,g)=\frac12\mathbf{D}(f,g)$ for $f,g\in H^1(\mathbb{R})$. Both $H^1(\mathbb{R})$ and $\mathcal{F}$ are Hilbert spaces under $\mathcal{E}_\alpha$ and hence there is $\alpha$-orthogonal compliment $\mathcal{G}_\alpha$. We give the explicit expression for functions in $\mathcal{G}_\alpha$ which then can be described by another two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the darning method, their regular representations are given.