Exponential contraction rates for a class of degenerate SDEs with Lévy noises

Abstract

Given a separable and real Hilbert space $H$, we consider the following stochastic differential equation (SDE) on $H$:$d{X}_t=-X_{t}dt+b\left(X_t\right)dt+d{Z}_t,$ where $Z:={\left(Z_t\right)}_{t\ge 0}$ is a cylindrical pure jump Lévy process on $H$ which may be degenerate in the sense that the support of Z is contained in a finite dimensional space. When the nonlinear drift term $b\left(x\right)$ is contractive with respect to some proper modified norm of $H$ for large distances, we obtain explicit exponential contraction rates of the SDE above in terms of Wasserstein distance under mild assumptions on the Lévy process Z. The approach is based on the refined basic coupling of Lévy noises, and it also works well when the so-called Lyapunov condition is satisfied.