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

Abstract

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 $$⟨\mathbf{b}(\mathbf{x})-\mathbf{b}(\mathbf{y}),\mathbf{x}-\mathbf{y}⟩⩽K_1|\mathbf{x}-\mathbf{y}{|}^2,\forall |\mathbf{x}-\mathbf{y}|⩽L,$$ $$⟨\mathbf{b}(\mathbf{x})-\mathbf{b}(\mathbf{y}),\mathbf{x}-\mathbf{y}⟩⩽-K_2|\mathbf{x}-\mathbf{y}{|}^{2+\theta },\forall |\mathbf{x}-\mathbf{y}|>L.$$ 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 $\theta ⩾0$. For the Lipschitz continuous function $h$, the normal CLT does not necessarily hold when $\theta =0$, but it is satisfied for $\theta >1-\frac{\alpha }{2}$.