Invariant measures for sdes driven by lévy noise: A case study for dissipative nonlinear drift in infinite dimension

We study a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Levy noise. We define a Hilbert-Banach setting in which we prove existence and uniqueness of solutions under general assumptions on the drift and the Levy noise. We then prove a decomposition of the solution process into a stationary component, the law of which is identified with the unique invariant probability measure mu of the process, and a component which vanishes asymptotically for large times in the L-P (mu)-sense, for all 1 <= p < + infinity.