Variance renormalisation of two dimensional gPAM with differentiated space white noise
Date:
In this work, we consider the gPAM equation with a differentiated space white noise in two dimensions. A particularity of this equation is that it just falls on the borderline of the subcriticality condition; moreover, even if one could have chosen a slightly more regular noise, the variance of certain non-linear functionals of the noise is expected to explode and the local solution theory would still fail. To tame down this variance blowup, a multiplicative renormalisation is introduced. This multiplicative renormalisation was first carried out by [Hairer ‘24] in the case of KPZ, and a general prediction was made there for a wider class of equations. The current work therefore has the objective to show gPAM falls into this picture. It is worth noting that, while a Da Prato-Debussche trick was used in the KPZ case, no such trick is available for gPAM. One thus has to work on the level of singular SPDE machineries such as regularity structure to obtain the desired result.
Based on joint work with Máté Gerencsér.