I have a question about semigroups of one-dimensional diffusions.

Let $X$ be the Ornstein Uhlenbeck process on $\mathbb{R}$. The generator is expresses as
$$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$
It is known that the $L^2$ semigroup $\{T_t\}$ of $X$ is a compact operator on $L^{2}(\mathbb{R},m)$, where $m$ is the speed measure of $X$. $\{T_t\}$ is extended to strongly continuous contraction semigroup on $L^{p}(\mathbb{R},m)$, $1\le p<\infty$. Moreover, $\{T_t\}$ becomes a compact operator on $L^{p}(\mathbb{R},m)$ for any $1<p<\infty$. $\{T_t\}$ is **not** a compact operator on $L^{1}(\mathbb{R},m)$.

**My question**

- Is there are nontrivial diffusion on $\mathbb{R}$ whose semigroup is compact on $L^{p}(\mathbb{R})$ for any $1 \le p \le \infty$? Consider the following differential operator: \begin{equation*} \frac{d^2}{dx^2}-x^{3}\frac{d}{dx}. \end{equation*} and the diffusion $Y$ associated with the above operator. The semigroups associated with $Y$ is compact on $L^{p}(\mathbb{R},m)$ for any $1 \le p \le \infty$? Here, $m$ is the speed measure of $Y$.

**Dirichlet form of $Y$**

The Dirichlet form of $Y$ is expresses as follows: \begin{align*} \mathcal{E}(f,g)&=\int_{\mathbb{R}}f'g'd\mu,\quad f,\ g \in \mathcal{F}\\ \mathcal{F}&=\{f \in L^{2}(\mathbb{R},\mu): f' \in L^{2}(\mathbb{R}.\mu)\}, \end{align*} where $d\mu=\exp(-x^4/2)\,dx$ and $dx$ is the Lebesgue measure on $\mathbb{R}$, and $f'$ is the distributional derivative of $f$.