Refined basic couplings and Wasserstein-type distances for SDEs with Levy noises

We establish the exponential convergence with respect to the L-1-Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equation d X-t = d Z(t) + b(X-t) dt, where (Z(t))(t >= 0) is a pure jump Levy process whose Levy measure v fulfills inf(x is an element of Rd, vertical bar x vertical bar <=kappa 0) [v boolean AND (delta(x) (*) nu)](R-d) 0 for some constant kappa(0) > 0, and the drift term b satisfies that for any x, y is an element of R-d, < b(x) - b(y), x - y > <= { Phi(1)(vertical bar x - y vertical bar) vertical bar x - y vertical bar, vertical bar x - y vertical bar < l(0); - K-2 vertical bar x - y vertical bar(2), vertical bar x - y vertical bar >= l(0) with some positive constants K-2, l(0) and positive measurable function Phi(1). The method is based on the refined basic coupling for Levy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process (X-t)(t >= 0). (C) 2018 Elsevier B.V. All rights reserved.