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.

Recommended citation: Gerencsér, Máté and Yueh-Sheng Hsu. “Variance renormalisation in regularity structures -- the case of $2d$ gPAM.” (2026).