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

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 $\mathcal{E}_α$ and hence there is $α$-orthogonal compliment $\mathcal{G}_α$. We give the explicit expression for functions in $\mathcal{G}_α$ 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.

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