  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.
