LAUSR

We consider a 2D stochastic modified Swift–Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space H2m for each m⩾1 . Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on H2m . At last, we improve the regularity of the obtained invariant measures to H2(m+1) . With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by Glatt-Holtz et al in their situation (2014 J. Math. Phys. 55 277–304).