Abstract
We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato-Debussche regime. The approach is essentially algebraic and involves using the generator equation of the SPDE at stationarity. Our results in particular cover the Φδ4 measures in dimensions δ < 14/5 , which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.