Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.13014 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916952157454336 |
|---|---|
| author | Jin, Xinghu Zhang, Xiaolong |
| author_facet | Jin, Xinghu Zhang, Xiaolong |
| contents | This paper establishes the quantitative stability of invariant measures $μ_α$ for $\mathbb{R}^d$-valued ergodic stochastic differential equations driven by rotationally invariant multiplicative $α$-stable processes with $α\in(1,2]$. Under structural assumptions on the coefficients with a fixed parameter vector $\bmθ$, we derive optimal convergence rates in the Wasserstein-$1$ ($\cW_{1}$) distance between the invariant measures introduced above, namely,
\item[(i)] For any interval $[α_0, \vartheta_0] \subset (1,2)$, there exists $C_1 = C(α_0, \vartheta_0,\bmθ,d) > 0$ such that
\cW_{1}(μ_α, μ_\vartheta) \leq C_1 |α- \vartheta|, \quad \forall α, \vartheta \in [α_0, \vartheta_0].
\item[(ii)] For any $α_0\in (1,2)$, there exists $C_2 = C(α_0, \bmθ) > 0$ such that \begin{align*}
\cW_{1}(μ_α, μ_2) \leq C_2\, d(2 - α), \quad \forall α\in [α_0, 2).
The optimality of these rates is rigorously verified by explicit calculations for the Ornstein-Uhlenbeck systems in \cite{Deng2023Optimal}. It is worth emphasizing that \cite{Deng2023Optimal} addressed only case (ii) under additive noise, whereas our analysis establishes results for both cases (i) and (ii) under multiplicative $α$-stable noise, employing fundamentally different analytical methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13014 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Rates for Ergodic SDEs Driven by Multiplicative $α$-Stable Processes in Wasserstein-1 distance Jin, Xinghu Zhang, Xiaolong Probability 60B10, 60G51, 60H07 This paper establishes the quantitative stability of invariant measures $μ_α$ for $\mathbb{R}^d$-valued ergodic stochastic differential equations driven by rotationally invariant multiplicative $α$-stable processes with $α\in(1,2]$. Under structural assumptions on the coefficients with a fixed parameter vector $\bmθ$, we derive optimal convergence rates in the Wasserstein-$1$ ($\cW_{1}$) distance between the invariant measures introduced above, namely, \item[(i)] For any interval $[α_0, \vartheta_0] \subset (1,2)$, there exists $C_1 = C(α_0, \vartheta_0,\bmθ,d) > 0$ such that \cW_{1}(μ_α, μ_\vartheta) \leq C_1 |α- \vartheta|, \quad \forall α, \vartheta \in [α_0, \vartheta_0]. \item[(ii)] For any $α_0\in (1,2)$, there exists $C_2 = C(α_0, \bmθ) > 0$ such that \begin{align*} \cW_{1}(μ_α, μ_2) \leq C_2\, d(2 - α), \quad \forall α\in [α_0, 2). The optimality of these rates is rigorously verified by explicit calculations for the Ornstein-Uhlenbeck systems in \cite{Deng2023Optimal}. It is worth emphasizing that \cite{Deng2023Optimal} addressed only case (ii) under additive noise, whereas our analysis establishes results for both cases (i) and (ii) under multiplicative $α$-stable noise, employing fundamentally different analytical methods. |
| title | Optimal Rates for Ergodic SDEs Driven by Multiplicative $α$-Stable Processes in Wasserstein-1 distance |
| topic | Probability 60B10, 60G51, 60H07 |
| url | https://arxiv.org/abs/2509.13014 |