Finite-Green-energy measure pairing with a Gaussian free field

Let $\rho$ be a finite measure on $D$ with finite Green energy, in the normalization of the Green-kernel expansion in the Dirichlet space. For an orthonormal basis $(e_n)$ and independent standard Gaussian random variables ${\xi }_n$, defineThe sum converges in L2 space, since the sum of the coefficient squares equals the Green energy. It is centered Gaussian withFor several measures, the same coefficients give their cross-covariances by the corresponding mixed Green integrals.

The joint law is independent of the chosen basis. A stronger pathwise statement needs consistent coefficients: realize the field through one isonormal Gaussian process $W$ on the Dirichlet Hilbert space and take ${\xi }_n=W(e_n)$ for each basis. Then Parseval's identity makes both expansions equal to the same $W$-pairing in L2 space. Drawing a fresh independent coefficient family for a different basis produces another field with the same law, not the same outcome-by-outcome field. This distinguishes basis invariance of a construction from an unspecified coupling of two constructions.^[ref]