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We are interested in the following two $\mathbb{R}^d$-valued stochasticdifferential equations (SDEs):  \begin{gather*}  d X_t=b(X_t)\, d t + \sigma\,d L_t, \quad X_0=x,  %\label{BM-SDE}  d Y_t=b(Y_t)\,dt + \sigma\,d B_t, \quad Y_0=y,  \end{gather*}  where $\sigma$ is an invertible $d\times d$ matrix, $L_t$ is a rotationallysymmetric $\alpha$-stable L\'evy process, and $B_t$ is a $d$-dimensionalstandard Brownian motion. We show that for any $\alpha_0 \in (1,2)$ theWasserstein-$1$ distance $W_1$ satisfies for $\alpha \in [\alpha_0,2)$  \begin{gather*}  W_{1}\left(X_{t}^x, Y_{t}^y\right)  \leq C e^{-Ct}|x-y|  +C_{\alpha_0}d\cdot\log(1+d)(2-\alpha)\log\frac{1}{2-\alpha},  \end{gather*}  which implies, in particular, \normal  \begin{equation} \label{e:W1Rate}  W_1(\mu_\alpha, \mu)  \leq C_{\alpha_0} d \cdot \log(1+d)(2-\alpha) \log \frac{1}{2-\alpha},  \end{equation}  where $\mu_\alpha$ and $\mu$ are the ergodic measures of $(X^x_t)_{t \ge 0}$and $(Y^y_t)_{t \ge 0}$ respectively.  The term $d\cdot\log(1+d)$ appearing in this estimate seems to be optimal.For the special case of a $d$-dimensional Ornstein--Uhlenbeck system, we showthat $W_1(\mu_\alpha, \mu) \geq C_{\alpha_0,d} (2-\alpha)$; this indicates thatthe convergence rate with respect to $\alpha$ in \eqref{e:W1Rate} is optimal upto a logarithmic correction. We conjecture that the sharp rate with respect to$\alpha$ and $d$ is $d\cdot\log (1+d) (2-\alpha)$.

Wasserstein-$1$ distance between SDEs driven by Brownian motion and stable processes