  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 renormalised
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
renormalised cubic nonlinearity.
