  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, $\alpha \in (0,2)$, introduced in Priola-Zabczyk
"Structural properties of semilinear SPDEs driven by cylindrical stable
processes" (PTRF to appear). 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.
