The generalised parabolic Anderson model (gPAM) is the paradigmatic example of a multiplicative singular SPDE. It has long been studied in an Itö sense when the driving noise is a Wiener process, but one of the early triumphs of path-wise approaches to singular SPDE, such as regularity structures and paracontrolled calculus, was a robust local well-posedness theory for gPAM that allows for a wider class of rough driving noises.
However, control over gPAM over large scales in these path-wise frameworks has been an open problem for quite some time. I will describe how the roughness of the noise at small scales makes control on the large scale behavior of gPAM very difficult, and also report on recent progress overcoming these difficulties. This is based on joint work with Guilherme Feltes and Hendrik Weber.