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 αα-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 in [8]. 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 [27], with Hölder continuous drift and a general, non-degenerate, symmetric αα-stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical αα-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 [28], with a different method.