Non-Gaussianity of invariant measures to SPDEs in Da Prato-Debussche regime

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 ${\mathrm{\Phi }}_{\delta }^4$ measures in dimensions $\delta <\frac{14}{5}$, which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.