Optimal Rates for Ergodic SDEs Driven by Multiplicative $\alpha$-Stable Processes in Wasserstein-1 distance

Abstract

This paper establishes the quantitative stability of invariant measures $\mu_{\alpha}$ for $\mathbb{R}^d$-valued ergodic stochastic differential equations driven by rotationally invariant multiplicative $\alpha$-stable processes with $\alpha\in(1,2]$. Under structural assumptions on the coefficients with a fixed parameter vector $\bm{\theta}$, we derive optimal convergence rates in the Wasserstein-$1$ ($\cW_{1}$) distance between the invariant measures introduced above, namely, \item[(i)] For any interval $[\alpha_0, \vartheta_0] \subset (1,2)$, there exists $C_1 = C(\alpha_0, \vartheta_0,\bm{\theta},d) > 0$ such that \cW_{1}(\mu_\alpha, \mu_\vartheta) \leq C_1 |\alpha - \vartheta|, \quad \forall \alpha, \vartheta \in [\alpha_0, \vartheta_0]. \item[(ii)] For any $\alpha_0\in (1,2)$, there exists $C_2 = C(\alpha_0, \bm{\theta}) > 0$ such that \begin{align*} \cW_{1}(\mu_\alpha, \mu_2) \leq C_2\, d(2 - \alpha), \quad \forall \alpha \in [\alpha_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 $\alpha$-stable noise, employing fundamentally different analytical methods.