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

The paper establishes, for the 2D stochastic modified Swift–Hohenberg equation du + [A^2u + f(u)]dt = φ(u)dW with f(u) = (a+3)u − 4Au + b|∇u|^2 + u^3, the existence of local and global martingale and pathwise solutions in H^{2m} (m>1), Feller continuity of the associated Markov semigroup, existence of invariant and ergodic invariant measures, and improved regularity of invariant measures up to H^{2(m+1)} and even C^∞ under stronger diffusion assumptions; the crucial smallness |b|<4 appears explicitly in their energy estimates and global arguments . The candidate solution reaches the same conclusions under the same structural conditions (A=-Δ+1, m>1, |b|<4, global Lipschitz φ at the appropriate Sobolev levels), using a closely related Galerkin/Itô/BDG framework plus an alternative local monotonicity framing for uniqueness/existence on a fixed basis. The only minor discrepancy is a non-substantive coefficient in the absorption of the b|∇u|^2 term; the paper’s detailed estimate shows the precise |b|/4 threshold at the L^2 level and propagates to higher levels, while the candidate’s inequality uses a comparable absorption constant (still requiring |b|<4). Overall, both arguments are valid and compatible; the paper uses Yamada–Watanabe and a diagonal argument for global martingale solutions, whereas the model mentions the locally monotone SPDE framework as an alternative path to pathwise solutions .