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

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 $(P_t{)}_{t\ge 0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in [1,\mathrm{\infty })$. In particular, consider the following SDE: $$\mathrm{d}{X}_t=\mathrm{d}{Z}_t+b(X_t)\mathrm{d}t,$$ where $(Z_t{)}_{t\ge 0}$ 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 $x,y\in {\mathbb{R}}^d$, $$⟨b(x)-b(y),x-y⟩\le \text{cases}{K}_1\mathrm{\mid }x-y{\mathrm{\mid }}^2,\mathrm{\mid }x-y\mathrm{\mid }\le L_0;\text{cr}-K_2\mathrm{\mid }x-y{\mathrm{\mid }}^{\theta },\mathrm{\mid }x-y\mathrm{\mid }>L_0$$ holds with some positive constants $K_1$, $K_2$, $L_0>0$ and $\theta \ge 2$, then there is a constant $\lambda :=\lambda (\theta ,K_1,K_2,L_0)>0$ such that for all $p\in [1,\mathrm{\infty })$, $t>0$ and $x,y\in {\mathbb{R}}^d$, $$W_p({\delta }_x{P}_t,{\delta }_y{P}_t)\le C(p,\theta ,K_1,K_2,L_0){\mathrm{e}}^{-\lambda t\mathrm{/}p}\left[\frac{\mathrm{\mid }x-y{\mathrm{\mid }}^{1\mathrm{/}p}\vee \mathrm{\mid }x-y\mathrm{\mid }}{1+\mathrm{\mid }x-y\mathrm{\mid }{\mathbf{1}}_{(1,\mathrm{\infty })\times (2,\mathrm{\infty })}(t,\theta )}\right]\mathrm{.}$$