Periodic space-time homogenisation of the $ϕ^4_2$ equation

Abstract

We consider the homogenisation problem for the $ϕ^4_2$ equation on the torus $\mathbb{T}^2$, namely the behaviour as $\varepsilon \to 0$ of the solutions to the equation suggestively written as $$ \partial_t u_\varepsilon - \nabla\cdot {A}(x/\varepsilon,t/\varepsilon^2) \nabla u_\varepsilon = -u^3_\varepsilon +ξ$$ where $ξ$ denotes space-time white noise and $A: \mathbb{T}^2\times \mathbb{R}$ is uniformly elliptic, periodic and Hölder continuous. When the noise is regularised at scale $δ\ll 1$ we show that any joint limit $\varepsilon,δ\to 0$ recovers the classical dynamical $ϕ^4_2$ 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 $A$.