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

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