Periodic space-time homogenisation of the ${\varphi }_2^4$ equation

Abstract

We consider the homogenisation problem for the ${\varphi }_2^4$ equation on the torus ${\mathbb{T}}^2$, namely the behaviour as $\epsilon \to 0$ of the solutions to the equation suggestively written as $${\partial }_t{u}_{\epsilon }-\nabla \cdot A(x/\epsilon ,t/{\epsilon }^2)\nabla u_{\epsilon }=-u_{\epsilon }^3+\xi$$ 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 $\epsilon ,\delta \to 0$ recovers the classical dynamical ${\varphi }_2^4$ 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 $A$.

Accepted version