Exponential Ergodicity and Regularity for Equations with Lévy Noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument [9]. Irreducibility 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 [28], with Hölder continuous drift and a general, non-degenerate, symmetric α-stable noise, and infinite dimensional parabolic systems, introduced in [31], with Lipschitz drift and cylindrical α-stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [29], with a different method.