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 Lévy noise. We define a Hilbert–Banach setting in which we prove existence and uniqueness of solutions under general assumptions on the drift and the Lévy 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 μ of the process, and a component which vanishes asymptotically for large times in the Lp(/mu)-sense, for all 1≤p<+∞