Exponential loss of memory for the dynamic ${\mathrm{\Phi }}_2^4$ with small noise

We consider the dynamic ${\mathrm{\Phi }}_2^4$ model (or stochastic Allen--Cahn equation) formally given by the SPDE $$({\partial }_t-\mathrm{\Delta }){\varphi }_{\epsilon }=-{\varphi }_{\epsilon }^3+{\varphi }_{\epsilon }+\sqrt{2\epsilon }\xi$$ where $\xi$ is a space-time white noise. When $\epsilon$ is small the solutions of the equation spend long time intervals in metastable states before reaching equilibrium. This phenomenon is known as metastability.

We discuss a coupling argument for solutions started from suitable initial conditions in space dimension 2 as a consequence of metastability. Such a result is already known in space dimension 1. The basic obstacle in our case is that classical solution theory for SPDEs is not applicable here since the non-linear term is ill-defined due to the irregularity of $\xi$. Hence a renormalization is required to compensate the divergences of the non-linear term. This destroys the "nice" structure of the non-linear term and a deeper analysis on the level of the so-called "remainder" term is required. An interesting application of such a coupling argument appears in the proof of the Eyring--Kramers law in the theory of metastability.