  We are interested in the following two $\mathbb{R}^d$-valued stochastic
differential 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)\,d t + \sigma\,d B_t, \quad Y_0=y,
  \end{gather*}
  where $\sigma$ is an invertible $d\times d$ matrix, $L_t$ is a rotationally
symmetric $\alpha$-stable L\'evy process, and $B_t$ is a $d$-dimensional
standard Brownian motion (note that $B_t$ is a rotationally symmetric
$\alpha$-stable L\'evy process with $\alpha=2$). We show that for any $\alpha_0
\in (1,2)$ the Wasserstein-$1$ distance $W_1$ satisfies for $\alpha \in
[\alpha_0,2)$
  \begin{gather*}
  W_{1}\left(X_{t}^x, Y_{t}^y\right)
  \leq C_1 e^{-C_2t}|x-y|
  +\frac{C}{\alpha_0-1}(2-\alpha)d\log(1+d),
  \end{gather*}
  which implies, in particular,
  \begin{equation} \label{e:W1Rate}
  W_1(\mu_\alpha, \mu_2)
  \leq \frac{C}{\alpha_0-1}(2-\alpha)d\log(1+d),
  \end{equation}
  where $\mu_\alpha$ and $\mu_2$ are the ergodic measures of $X_t$ and $Y_t$
respectively.
  For the special case of a $d$-dimensional Ornstein--Uhlenbeck system, we show
that $W_1(\mu_\alpha, \mu_2) \geq C_{d} (2-\alpha)$ for all $\alpha\in(1,2)$;
this indicates that the convergence rate with respect to $\alpha$ in the second
bound is optimal. The term $d\log(1+d)$ appearing in this bound seems to be
optimal for the dimension $d$ as well.
