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

Abstract

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.