$L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes

Abstract

Coupling by reflection mixed with synchronous coupling is constructed for a class of stochastic differential equations (SDEs) driven by Lévy 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 $x,y\in\mathbb{R}^{d}$,

\[\langle b(x)-b(y),x-y\rangle\le\cases{K_{1}|x-y|^{2},\quad\phantom{-} |x-y|\le L_{0};\cr-K_{2}|x-y|^{\theta},\quad |x-y|>L_{0}}\] 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 $x,y\in\mathbb{R}^{d}$,

\[W_{p}(\delta_{x}P_{t},\delta_{y}P_{t})\le C(p,\theta,K_{1},K_{2},L_{0})\mathrm{e}^{-\lambda t/p}[\frac{|x-y|^{1/p}\vee|x-y|}{1+|x-y|{\mathbf{1}}_{(1,\infty)\times(2,\infty)}(t,\theta)}].\]