Singularity of solutions to singular SPDEs

Building on the notes [Hai17], we give a sufficient condition for the marginal distribution of the solution of singular SPDEs on the $d$-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearised equation. As applications we obtain the singularity of the $\Phi^4_3$-measure with respect to the Gaussian free field measure and the border of parameters for the fractional $\Phi^4$-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.