Exponential ergodicity and regularity for equations with L\'evy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by $\alpha$-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Harris theorem, and the second on Doeblin's coupling argument. Irreducibility, Lyapunov function techniques, and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite-dimensional equations, introduced in [Priola 2010], with H\"older continuous drift and a general, non-degenerate, symmetric $\alpha$-stable noise, and infinite-dimensional parabolic systems, introduced in [Priola-Zabczyk 2009], with Lipschitz drift and cylindrical $\alpha$-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. %under the total variation norm. This improves, in particular, an earlier result established in [Priola-Xu-Zabczyk 2010] using the weak convergence induced by the Kantorovich-Wasserstein metric.