Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift–Hohenberg equation

Abstract

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 for each . 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 . At last, we improve the regularity of the obtained invariant measures to . 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).