Solution
= Solution
The continuum <Gaussian free field> $\Gamma$ on $\mathbb R^3$ is the centered Gaussian random distribution indexed by smooth compactly supported test functions, with covariance
$$
\mathbb E[\Gamma(f)\Gamma(g)]
=\iint_{\mathbb R^3\times\mathbb R^3}
f(x)G(x,y)g(y)dxdy,
\qquad
G(x,y)=\frac{c_3}{|x-y|},
$$
where $G$ is the Green kernel of $-\Delta$. Equivalently, its covariance operator is $(-\Delta)^{-1}$.