Periodic space-time homogenisation of the $φ^4_2$ 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 \textit{suggestively} written 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.
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