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

Abstract

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.