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

TL;DR

This paper investigates the Wasserstein-1 distance between solutions of SDEs driven by Brownian motion and stable processes, establishing bounds that quantify how the ergodic measures converge as the stability parameter approaches 2.

Contribution

It provides explicit bounds on Wasserstein-1 distances between SDE solutions driven by Brownian motion and stable processes, including optimal convergence rates in the stability parameter.

Findings

W_1 distance between solutions decays exponentially over time.

Explicit bounds on the ergodic measures' Wasserstein-1 distance as alpha approaches 2.

Optimal convergence rate with respect to the stability parameter alpha.

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.