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

Máté Gerencsér; Yueh-Sheng Hsu

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

Máté Gerencsér, Yueh-Sheng Hsu

Abstract

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.

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

Abstract

We consider the variance renormalisation of a singular SPDE for which a Da Prato-Debussche trick is not applicable. The example taken is the

-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

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.