Ergodicity of the Anderson $Φ^4_2$ model

We consider the parabolic stochastic quantization equation associated to the $\Phi^4_2$ model on the torus in a spatial white noise environment. We study the long time behavior of this heat equation with independent multiplicative white noise and additive spacetime white noise, which is a singular SPDE in a singular environement and requires two different renormalization procedures. We prove that the solution is global in time with a strong a priori $L^p$ bound independent of the initial data in $\mathcal{C}^{-\varepsilon}$ for large $p$ . The quenched solution given the environment is shown to be an infinite dimensional Markov process which satisfies the strong Feller property. We prove exponential convergence to a unique invariant measure using a Doeblin criterion for the transition semigroup. In particular, our work is a generalization of a previous work by Tsatsoulis and Weber in a case which is not translation invariant hence the method makes no use of the reversibility of the dynamics or the explicit knowledge of the invariant measure and it is therefore in principle applicable to situations where these are not available such as the vector-valued case.