### 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 E1×E2${E_{1}\times E_{2}}$ and expressed as Xt=(Xt1,XAt2),t≥0,$X_{t}=(X^{1}_{t},X^{2}_{A_{t}}),\quad t\geq 0,$ where Xi${X^{i}}$ is a symmetric diffusion on Ei${E_{i}}$ for i=1,2${i=1,2}$, and A is a positive continuous additive functional of X1${X^{1}}$. One of our main results indicates that any skew product type regular subspace of X , say Yt=(Yt1,YA~t2),t≥0,$Y_{t}=(Y^{1}_{t},{Y^{2}_{\tilde{A}_{t}}}),\quad t\geq 0,$ can be characterized as follows: the associated smooth measure of A~${\tilde{A}}$ is equal to that of A , and Yi${Y^{i}}$ corresponds to a regular subspace of Xi${X^{i}}$ for i=1,2${i=1,2}$. Furthermore, we shall make some discussions on rotationally invariant diffusions on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$, which are special skew product diffusions on (0,∞)×Sd-1${(0,\infty)\times S^{d-1}}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ to a new regular Dirichlet form on ℝd${\mathbb{R}^{d}}$. More precisely, fix a regular Dirichlet form (ℰ,ℱ)${(\mathcal{E,F}\kern 0.569055pt)}$ on the state space ℝd${\mathbb{R}^{d}}$. Its part Dirichlet form on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is denoted by (ℰ0,ℱ)0${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. Let (ℰ~0,ℱ~)0${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ be a regular subspace of (ℰ0,ℱ)0${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. We want to find a regular subspace (ℰ~,ℱ~)${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ of (ℰ,ℱ)${(\mathcal{E,F}\kern 0.569055pt)}$ such that the part Dirichlet form of (ℰ~,ℱ~)${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is exactly (ℰ~0,ℱ~)0${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. If (ℰ~,ℱ~)${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ exists, we call it a regular extension of (ℰ~0,ℱ~)0${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of (ℰ0,ℱ)0${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ has a unique regular extension.