Periodic homogenisation for two dimensional generalised parabolic Anderson model

We consider the periodic homogenisation problem for the generalised parabolic Anderson model on the two dimensional torus. We show that, for the renormalisation that respects Wick ordering, the homogenisation and renormalisation procedures commute. The main novelty is to identify a suitable solution ansatz beyond the usual para-controlled ansatz to set up a fixed point problem uniform in the homogenisation parameter. After that, one further utilises cancellations and resonances from the homogenisation oscillations to show convergences of both the solution and flux to the right limits. At a technical level, we frequently use integration by parts as well as "completing the products" to circumvent the incompatibility between para-products and variable coefficients. As a byproduct, we also show that the standard two dimensional generalised parabolic Anderson model can be constructed with para-controlled calculus without using commutator estimates.