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

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.