Linear Evolution Equations with Cylindrical L\'evy Noise: Gradient Estimates and Exponential Ergodicity

Explicit coupling property and gradient estimates are investigated for the linear evolution equations on Hilbert spaces driven by an additive cylindrical L\'evy process. The results are efficiently applied to establish the exponential ergodicity for the associated transition semigroups. In particular, our results extend recent developments on related topic for cylindrical symmetric $\alpha$-stable processes.