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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2504.21430 |
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- 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 $α$-stable process with $α\in(1,2)$ and $\mathbf{b}:\mathbb{R}^d \rightarrow \mathbb{R}^d$ is a differentiable function satisfying the following condition: there exist some $θ\ge 0$, and $K_1 , K_2 , L>0$, so that $$\langle \mathbf{b}(\mathbf{x})-\mathbf{b}(\mathbf{y}), \mathbf{x}-\mathbf{y}\rangle \leqslant K_1 |\mathbf{x}-\mathbf{y}|^2, \ \ \forall \ \ |\mathbf{x}-\mathbf{y}| \leqslant L, $$ $$\langle \mathbf{b}(\mathbf{x})-\mathbf{b}(\mathbf{y}), \mathbf{x}-\mathbf{y}\rangle \leqslant -K_2 |\mathbf{x}-\mathbf{y}|^{2+θ}, \ \ \forall \ \ |\mathbf{x}-\mathbf{y}| > L.$$ Under this assumption, the SDE admits a unique invariant measure $μ$. We investigate the normal central limit theorem (CLT) of the empirical measures $$ \mathcal{E}_t^\mathbf{x}(\cdot)=\frac{1}{t} \int_0^t δ_{\mathbf{X}_s }(\cdot) \mathrm{d} s, \ \ \ \ \mathbf{X}_0=\mathbf{x} \in \mathbb{R}^d, \ \ t>0, $$ where $δ_{\mathbf{x}}(\cdot)$ is the Dirac delta measure. Our results reveal that, for the bounded measurable function $h$, $$\sqrt t \left(\mathcal{E}_t^\mathbf{x}(h)-μ(h)\right)=\frac{1}{\sqrt t} \int_0^t \left(h\left(\mathbf{X}_s^\mathbf{x}\right)-μ(h)\right) \mathrm{d} s$$ admits a normal CLT for $θ\geqslant 0$. For the Lipschitz continuous function $h$, the normal CLT does not necessarily hold when $θ=0$, but it is satisfied for $θ>1-\fracα{2}$.