Invariant Gibbs measure for Anderson nonlinear wave equation

We study the Gaussian measure whose covariance is related to the Anderson Hamiltonian operator, proving that it admits a regular coupling to the (standard) Gaussian free field exploiting the stochastic optimal control formulation of Gibbs measures. Using this coupling, we define the renormalized powers of the Anderson free field and we prove that the associated quartic Gibbs measure is invariant under the flow of a nonlinear wave equation with renormalized cubic nonlinearity.