Gaussian fluctuations of differential observables of stationary lattice fields

We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete $d$-dimensional torus of mesh $1/N$. Our main result is a Central Limit Theorem for the fields $f\mapsto N^{-d/2-m}\sum_xϕ_x^N(Pf)(x/N)$, where $P$ is a scalar homogeneous constant-coeffic