Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

We study exponential ergodicity of a broad class of stochastic processes whose dynamics are governed by pure jump Lévy noise. In the first part of the paper we focus on solutions of stochastic differential equations (SDEs) whose drifts satisfy general Lyapunov-type conditions. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions, both in the total variation and Wasserstein distances. The second part of the paper is devoted to SDEs of McKean-Vlasov type with distribution dependent drifts. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. Then, extending our techniques from the first part of the paper, we obtain results on exponential ergodicity of solutions of McKean-Vlasov SDEs, under general conditions on the drift and the driving noise.