Periodic space-time homogenisation of the φ₂<SUP>4</SUP> equation

We consider the homogenisation problem for the φ42 equation on the torus T2, namely the behaviour as ε →0 of the solutions to the equation suggestivelywritten as ∂tuε − ∇ · A(x/ε, t/ε2)∇uε = −u3ε + ξ where ξ denotes space-time white noise and A :T2×R is uniformly elliptic, periodic and Hölder continuous. When the noise is regularised at scale δ < 1 we show that any joint limit ε, δ→0 recovers the classical dynamical φ42 model. In certain regimes or if the regularisation is chosen in a specific way adapted to the problem, we show that the counterterms can be chosen as explicit local functions of A.