Explicit invariant measures for infinite dimensional SDE driven by L\'evy noise with dissipative nonlinear drift I

We stu\dd y a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by L\'evy noise. Our work is divided in two parts. In the present part I we first define a Hilbert-Banach setting in which we can prove existence and uniqueness of solutions under general assumptions on the drift and the L\'evy noise. We then prove a decomposition of the solution process in a stationary component and a component which vanishes asymptotically for large times in the $L^p-$sense, $p\geq1$. The law of the stationary component is identified with the unique invariant probability measure of the process. In part II we will exhibit the invariant measure as the limit of explicit invariant measures for finite dimensional approximants.