Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 4 / d / Solution

Choose an orthonormal basis $(e_n)$ of $H_0^1(D)$ and write the GFF formally as $h={\sum }_n{\xi }_n{e}_n$, where the ${\xi }_n$ are independent standard normal variables. The Green-kernel expansion in the Dirichlet space givesConsequently the seriesconverges in $L^2$. Its partial sums are centered Gaussian and their variances converge to the displayed Green energy. The $L^2$ limit is therefore Gaussian with mean zero and varianceThis constructs the Finite-Green-energy measure pairing with a Gaussian free field independently of the chosen orthonormal basis.