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

arXiv2302.03372MaRDI QIDQ6508722

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Abstract: We are interested in the following two $mathbb{R}^d$-valued stochastic differential equations (SDEs): �egin{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 $dimesd$ 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 $alph{a}_{0}in(1,2)$ the Wasserstein-$1$ distance $W_1$ satisfies for $alphain[alph{a}_0,2)$ �egin{gather*} W_{1}left(X_{t}^x, Y_{t}^y ight) leq C e^{-Ct}|x-y| +C_{alpha_0}dcdotlog(1+d)(2-alpha)logfrac{1}{2-alpha}, end{gather*} which implies, in particular, ormal �egin{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 $m{u}_{a}lpha$ and $mu$ are the ergodic measures of $(X_t^x{)}_{tge0}$ and $(Y_t^y{)}_{tge0}$ respectively. The term $dcdotlog(1+d)$ appearing in this estimate seems to be optimal. For the special case of a $d$-dimensional Ornstein--Uhlenbeck system, we show that $W_1(m{u}_{a}lpha,mu)geq{C}_{alph{a}_0,d}(2-alpha)$; 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 $dcdotlog(1+d)(2-alpha)$.

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