$L^p$-Wasserstein distance for stochastic differential equations driven by L\'{e}vy processes

Abstract

Coupling by reflection mixed with synchronous coupling is constructed for a class of stochastic differential equations (SDEs) driven by L\'{e}vy noises. As an application, we establish the exponential contractivity of the associated semigroups $(P_t)_{t\ge0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in[1,\infty)$. In particular, consider the following SDE: \[\mathrm{d}X_t=\mathrm{d}Z_t+b(X_t)\,\mathrm{d}t,\] where $(Z_t)_{t\ge0}$ is a symmetric $\alpha$-stable process on $\mathbb{R}^d$ with $\alpha\in(1,2)$. We show that if the drift term $b$ satisfies that for any $x,y\in\mathbb{R}^d$, \[\bigl\langle b(x)-b(y),x-y\bigr\rangle\le\cases{K_1|x-y|^2,\qquad |x-y|\le L_0;\cr -K_2|x-y|^{\theta},\qquad |x-y|>L_0}\] holds with some positive constants $K_1$, $K_2$, $L_0>0$ and $\theta\ge2$, then there is a constant $\lambda:=\lambda(\theta,K_1,K_2,L_0)>0$ such that for all $p\in[1,\infty)$, $t>0$ and $x,y\in\mathbb{R}^d$, \[W_p(\delta_xP_t,\delta_yP_t)\le C(p,\theta,K_1,K_2,L_0)\mathrm{e}^{-\lambda t/p}\biggl[\frac{|x-y|^{1/p}\vee|x-y|}{1+|x-y|{\mathbf{1}}_{(1,\infty )\times (2,\infty)}(t,\theta)}\biggr].\]