Variance renormalisation in regularity structures -- the case of $2d$ gPAM

We consider the variance renormalisation of a singular SPDE for which a Da Prato-Debussche trick is not applicable. The example taken is the $2$-dimensional generalised parabolic Anderson model (gPAM), driven by a much rougher than white noise, necessitating both a multiplicative and an additive renormalisation. To handle the discrepancy between the regularity structures of the approximate and the limiting equations, we consider models that lift $0$ noises to nontrivial models, in analogy with ``pure area’’ from rough paths. The convergence to such a model is shown for the BPHZ model over the vanishing noise via graphical computations.