Regular subspaces of skew product diffusions

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 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 $$X_t=(X^1_t,X^2_{A_t}),\quad t\geq 0,$$ 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 $$Y_t=(Y^1_t,Y^2_{\tilde{A}_t}),\quad t\geq 0,$$ can be characterized as follows: the associated smooth measure of $\tilde{A}$ 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 $\mathbf{R}^d\setminus \{0\}$, which are special skew product diffusions on $(0,\infty)\times S^{d-1}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on $\mathbf{R}^d\setminus \{0\}$ to a new regular Dirichlet form on $\mathbf{R}^d$.