For a stochastic porous media lattice system driven by infinitely dimensional nonlinear noise, we investigate its invariant measures and mean dynamics on the enlarged optimized \(l^p\) -space for \(p>2\) rather than the usual \(l^2\) -space in the literature, where the former space contains the latter. We prove existence and uniqueness of enlarged solutions as well as enlarged mean weak attractors. We then show existence of an enlarged invariant measure. We finally establish upper semicontinuity of the set of all enlarged invariant measures in noise density. It is the first time to deduce invariant measures for stochastic lattice models when the underlying space is a Banach space rather than a Hilbert space, while Itô’s formula for the higher power function is applied to prove the asymptotic tail property of enlarged solutions and asymptotic tightness of enlarged distributions.
