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 $\mathbf{ b}:\mathbb{ R}^d \rightarrow \mathbb{ R}^d$ is a differentiable function satisfying the following condition: there exist some $\theta \ge 0$, and $K_1 , K_2 , L>0$, so that $$\langle \mathbf{ b}(\mathbf{ x})-\mathbf{ b}(\mathbf{ y}), \mathbf{ x}-\mathbf{ y}\rangle \leqslant K_1 |\mathbf{ x}-\mathbf{ y}|^2, \ \ \forall \ \ |\mathbf{ x}-\mathbf{ y}| \leqslant L, $$ $$\langle \mathbf{ b}(\mathbf{ x})-\mathbf{ b}(\mathbf{ y}), \mathbf{ x}-\mathbf{ y}\rangle \leqslant -K_2 |\mathbf{ x}-\mathbf{ y}|^{ 2+\theta}, \ \ \forall \ \ |\mathbf{ x}-\mathbf{ y}| > L.$$ 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 $h$, $$\sqrt t \left(\mathcal{ E}_t^\mathbf{ x}(h)-\mu(h)\right)=\frac{ 1}{ \sqrt t} \int_0^t \left(h\left(\mathbf{ X}_s^\mathbf{ x}\right)-\mu(h)\right) \mathrm{ d} s$$ admits a normal CLT for $\theta \geqslant 0$. For the Lipschitz continuous function $h$, the normal CLT does not necessarily hold when $\theta=0$, but it is satisfied for $\theta>1-\frac{ \alpha}{ 2}$.
Categories: math.PR