  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 $H^{2m}$ for each $m\geqslant1$. 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 $H^{2m}$. At last, we improve the regularity of the
obtained invariant measures to $H^{2(m+1)}$. With appropriate conditions on the
diffusion coefficient, we can deduce the infinite regularity of the invariant
measures, which was conjectured by Glatt-Holtz \textit{et al.} in their
situation.
