Existence and non-existence of the CLT for a family of SDEs driven by stable process
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| Format: | Preprint |
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2025
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| _version_ | 1866910922827628544 |
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| author | Mo, Yingjun Wang, Yu |
| author_facet | Mo, Yingjun Wang, Yu |
| contents | 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 $θ>1-\fracα{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21430 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and non-existence of the CLT for a family of SDEs driven by stable process Mo, Yingjun Wang, Yu Probability 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 $θ>1-\fracα{2}$. |
| title | Existence and non-existence of the CLT for a family of SDEs driven by stable process |
| topic | Probability |
| url | https://arxiv.org/abs/2504.21430 |