$L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.3150/15-bej705
· OA: W2298218077
Coupling by reflection mixed with synchronous coupling is constructed for a\nclass of stochastic differential equations (SDEs) driven by L\\'{e}vy noises. As\nan application, we establish the exponential contractivity of the associated\nsemigroups $(P_t)_{t\\ge0}$ with respect to the standard $L^p$-Wasserstein\ndistance for all $p\\in[1,\\infty)$. In particular, consider the following SDE:\n\\[\\mathrm{d}X_t=\\mathrm{d}Z_t+b(X_t)\\,\\mathrm{d}t,\\] where $(Z_t)_{t\\ge0}$ is a\nsymmetric $\\alpha$-stable process on $\\mathbb{R}^d$ with $\\alpha\\in(1,2)$. We\nshow that if the drift term $b$ satisfies that for any $x,y\\in\\mathbb{R}^d$,\n\\[\\bigl\\langle b(x)-b(y),x-y\\bigr\\rangle\\le\\cases{K_1|x-y|^2,\\qquad |x-y|\\le\nL_0;\\cr -K_2|x-y|^{\\theta},\\qquad |x-y|>L_0}\\] holds with some positive\nconstants $K_1$, $K_2$, $L_0>0$ and $\\theta\\ge2$, then there is a constant\n$\\lambda:=\\lambda(\\theta,K_1,K_2,L_0)>0$ such that for all $p\\in[1,\\infty)$,\n$t>0$ and $x,y\\in\\mathbb{R}^d$, \\[W_p(\\delta_xP_t,\\delta_yP_t)\\le\nC(p,\\theta,K_1,K_2,L_0)\\mathrm{e}^{-\\lambda\nt/p}\\biggl[\\frac{|x-y|^{1/p}\\vee|x-y|}{1+|x-y|{\\mathbf{1}}_{(1,\\infty )\\times\n(2,\\infty)}(t,\\theta)}\\biggr].\\]\n
