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

Changsong Deng, Rene L. Schilling, Lihu Xu

Published 2023-02-07Version 1

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)\,dt + \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. 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 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 show that $W_1(\mu_\alpha, \mu) \geq C_{\alpha_0,d} (2-\alpha)$; this indicates that the convergence rate with respect to $\alpha$ in \eqref{e:W1Rate} is optimal up to a logarithmic correction. We conjecture that the sharp rate with respect to $\alpha$ and $d$ is $d\cdot\log (1+d) (2-\alpha)$.