Lp-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 (Pt )t≥0 with respect to the standard Lp-Wasserstein distance for all p ϵ [1,∞). In particular, consider the following SDE: dXt = dZt +b(Xt) dt, where (Zt )t≥0 is a symmetric α-stable process on Rd with α ϵ (1, 2). We show that if the drift term b satisfies that for any x,y ϵ Rd , b(x) -b(y), x -y ≤ K1|x -y|2, |x - y| ≤ L0; -K2|x - y|θ , |x - y|>L0 holds with some positive constants K1, K2, L0 > 0 and θ ≥ 2, then there is a constant λ := λ(θ,K1,K2, L0) > 0 such that for all p ϵ [1,∞), t > 0 and x,y ϵ Rd, Wp(δxPt , δyPt) ≤ C(p, θ,K1,K2,L0)e-λt/p|x -y|1/p V |x -y| 1 + |x -y|1(1,∞)×(2,∞)(t, θ).