Test-function pairing with a Gaussian free field
= Test-function pairing with a Gaussian free field
{c}
For a <test function> $\phi$ on $D$, let $f_\phi=-2\pi\Delta^{-1}\phi\in H_0^1(D)$. The pairing
$$
(h,\phi):=(h,f_\phi)_\nabla
$$
is a centered <Gaussian random variable> with variance
$$
\iint_{D\times D}\phi(x)G_D(x,y)\phi(y)\,dx\,dy.
$$