Existence and non-existence of the CLT for a family of SDEs driven by stable process

Stochastic differential equations (SDEs) without global Lipschitz drift often demonstrate unusual phenomena. In this paper, we consider the following SDE on ${\mathbb{R}}^d$: \begin{align*} $\mathrm{d}$ $\mathbf{X}$_t=$\mathbf{b}$($\mathbf{X}$_t) $\mathrm{d}$ t+ $\mathrm{d}$$\mathbf{Z}$_t, \quad $\mathbf{X}$_0=$\mathbf{x}$ \in $\mathbb{R}$^d, \end{align*} where ${\mathbf{Z}}_t$ is the rotationally symmetric $\alpha$-stable process with $\alpha \in (1,2)$ and $\mathbf{b}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is a differentiable function satisfying the following condition: there exist some $\theta \ge 0$, and $K_1,K_2,L>0$, so that $$⟨\mathbf{b}(\mathbf{x})-\mathbf{b}(\mathbf{y}),\mathbf{x}-\mathbf{y}⟩⩽K_1\mathrm{\mid }\mathbf{x}-\mathbf{y}{\mathrm{\mid }}^2,\mathrm{\forall }\mathrm{\mid }\mathbf{x}-\mathbf{y}\mathrm{\mid }⩽L,$$ $$⟨\mathbf{b}(\mathbf{x})-\mathbf{b}(\mathbf{y}),\mathbf{x}-\mathbf{y}⟩⩽-K_2\mathrm{\mid }\mathbf{x}-\mathbf{y}{\mathrm{\mid }}^{2+\theta },\mathrm{\forall }\mathrm{\mid }\mathbf{x}-\mathbf{y}\mathrm{\mid }>L\mathrm{.}$$ Under this assumption, the SDE admits a unique invariant measure $\mu$. We investigate the normal central limit theorem (CLT) of the empirical measures $${\mathcal{E}}_t^{\mathbf{x}}(\cdot )=\frac{1}{t}{\int }_0^t{\delta }_{{\mathbf{X}}_s}(\cdot )\mathrm{d}s,{\mathbf{X}}_0=\mathbf{x}\in {\mathbb{R}}^d,t>0,$$ where ${\delta }_{\mathbf{x}}(\cdot )$ is the Dirac delta measure. Our results reveal that, for the bounded measurable function $h$, $$\sqrt{t}({\mathcal{E}}_t^{\mathbf{x}}(h)-\mu (h))=\frac{1}{\sqrt{t}}{\int }_0^t(h\left({\mathbf{X}}_s^{\mathbf{x}}\right)-\mu (h))\mathrm{d}s$$ admits a normal CLT for $\theta ⩾0$. For the Lipschitz continuous function $h$, the normal CLT does not necessarily hold when $\theta =0$, but it is satisfied for $\theta >1-\frac{\alpha }{2}$.