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

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 $$\Phi ^4_\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Φ</mml:mi> <mml:mi>δ</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> measures in dimensions $$\delta &lt;\frac{14}{5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mfrac> <mml:mn>14</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> , which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.