Exponential mixing for some SPDEs with Levy noise

We show how gradient estimates for transition semigroups
can be used to establish exponential mixing for a class of Markov
processes in infinite dimensions. We concentrate on semilinear
systems driven by cylindrical $\alpha$-stable noises. We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are sufficiently small.