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

Martin Hairer, Harprit Singh

Martin Hairer, Harprit Singh

**27/11/23**Published in : arXiv:2311.15788

We consider the homogenisation problem for the \phi^4_2 equation on the torus \mathbb{T}^2, namely the behaviour as \varepsilon \to 0 of the solutions to the equation \textit{suggestively} written as \partial_t u_\varepsilon - \nabla\cdot {A}(x/\varepsilon,t/\varepsilon^2) \nabla u_\varepsilon = -u^3_\varepsilon +\xi where \xi 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 \delta \ll 1 we show that any joint limit \varepsilon,\delta \to 0 recovers the classical dynamical \phi^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.