Let $d$ be an integer. We denote by $m$ the Lebesgue measure on $\mathbb{R}^d$. We define $BL(\mathbb{R}^d)$ by \begin{align*} BL(\mathbb{R}^d)=\{f \in L^2_{\rm loc}(\mathbb{R}^d,m) \mid |\nabla f|\in L^2(\mathbb{R}^d,m)\}, \end{align*} where $L^2_{\rm loc}(\mathbb{R}^d,m)$ denotes the space of locally square integrable function on $\mathbb{R}^d$. For $f \in L^2_{\rm loc}(\mathbb{R}^d,m)$, we write $\nabla f$ for the distributional derivative. For $f,g \in BL(\mathbb{R}^d)$, we define $\mathcal{E}(f,g)=\int_{\mathbb{R}^d}\nabla f \cdot \nabla g\,dm$, where $\cdot$ denotes the standard inner product on $\mathbb{R}^d$.

We take a positive continuous function $V\colon \mathbb{R}^d \to \mathbb{R}$,which may be unbounded. We set \begin{align*} \mathcal{F}=\left\{f \in BL(\mathbb{R}^d) : \int_{\mathbb{R}^d}f^2\,Vdm<\infty \right\} \end{align*}

We can show that $(\mathcal{E},\mathcal{F})$ becomes a Dirichlet form on $L^2(\mathbb{R}^d,V\,dm)$. Therefore, $(\mathcal{E},\mathcal{F})$ generates a strongly continuous contraction semigroup $\{T_t\}_{t>0}$ on $L^2(\mathbb{R}^d,V\,dm)$, which is extended to a contraction semigroup on $L^{\infty}(\mathbb{R}^d,V\,dm)$. The extension is still denoted as $\{T_t\}_{t>0}$. In fact, $\{T_t\}_{t>0}$ is identified with the semigroup of a time-changed Brownian motion, and we can show that $\{T_t\}_{t>0}$ is a strongly continuous contraction semigroup on $C_{0}(\mathbb{R}^d)$. Here, $C_{0}(\mathbb{R}^d)$ stands for the space of continuous functions vanishing at infinity.

**My question**

Can we show that $\{T_t\}_{t>0}$ is extended to an bounded analytic semigroup on $C_{0}(\mathbb{R}^d)$? The generator of $\{T_t\}_{t>0}$ is given by $\frac{1}{V}\Delta$, where $\Delta$ is the Laplacian on $\mathbb{R}^d$.

I am interested in whether analyticity of semigroups are stable under time change, which is one of the most fundamental transformations of stochastic processes.