## Abstract

Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain.
What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions.
A skew product diffusion *X* is a symmetric Markov process on the product state space ${E}_{1}\times {E}_{2}$ and expressed as

${X}_{t}=({X}_{t}^{1},{X}_{{A}_{t}}^{2}),t\ge 0,$

where ${X}^{i}$ is a symmetric diffusion on ${E}_{i}$ for $i=1,2$, and *A* is a positive continuous additive functional of ${X}^{1}$.
One of our main results indicates that any skew product type regular subspace of *X*, say

${Y}_{t}=({Y}_{t}^{1},{Y}_{{\stackrel{~}{A}}_{t}}^{2}),t\ge 0,$

can be characterized as follows: the associated smooth measure of $\stackrel{~}{A}$ is equal to that of *A*, and ${Y}^{i}$ corresponds to a regular subspace of ${X}^{i}$ for $i=1,2$.
Furthermore, we shall make some discussions on rotationally invariant diffusions on ${\mathbb{R}}^{d}\setminus \{\mathrm{}\}$,
which are special skew product diffusions on $(0,\mathrm{\infty})\times {S}^{d-1}$.
Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ${\mathbb{R}}^{d}\setminus \{\mathrm{}\}$ to a new regular Dirichlet form on ${\mathbb{R}}^{d}$.
More precisely, fix a regular Dirichlet form $(\mathcal{\mathcal{E}},\mathcal{\mathcal{F}})$ on the state space ${\mathbb{R}}^{d}$.
Its part Dirichlet form on ${\mathbb{R}}^{d}\setminus \{\mathrm{}\}$ is denoted by $({\mathcal{\mathcal{E}}}^{0},\mathcal{\mathcal{F}}{}^{0})$.
Let $({\stackrel{~}{\mathcal{\mathcal{E}}}}^{0},\stackrel{~}{\mathcal{\mathcal{F}}}{}^{0})$ be a regular subspace of $({\mathcal{\mathcal{E}}}^{0},\mathcal{\mathcal{F}}{}^{0})$.
We want to find a regular subspace $(\stackrel{~}{\mathcal{\mathcal{E}}},\stackrel{~}{\mathcal{\mathcal{F}}})$ of $(\mathcal{\mathcal{E}},\mathcal{\mathcal{F}})$ such that the part Dirichlet form of $(\stackrel{~}{\mathcal{\mathcal{E}}},\stackrel{~}{\mathcal{\mathcal{F}}})$ on ${\mathbb{R}}^{d}\setminus \{\mathrm{}\}$ is exactly $({\stackrel{~}{\mathcal{\mathcal{E}}}}^{0},\stackrel{~}{\mathcal{\mathcal{F}}}{}^{0})$.
If $(\stackrel{~}{\mathcal{\mathcal{E}}},\stackrel{~}{\mathcal{\mathcal{F}}})$ exists, we call it a regular extension of $({\stackrel{~}{\mathcal{\mathcal{E}}}}^{0},\stackrel{~}{\mathcal{\mathcal{F}}}{}^{0})$. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of $({\mathcal{\mathcal{E}}}^{0},\mathcal{\mathcal{F}}{}^{0})$ has a unique regular extension.

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