Green-kernel expansion in the Dirichlet space

For an orthonormal basis $(e_n)$ of $H_0^1(D)$, the zero-Dirichlet Green function of the Laplacian is the weak kernel represented byConsequently, whenever the terms are defined and the right side is finite,

If a measure $\rho$ has finite Green energy, thenconverges in $L^2$ for any orthonormal basis $(e_n)$ of $H_0^1(D)$ and independent standard normal coefficients ${\xi }_n$. The limit is centered Gaussian with variance equal to the Green energy of $\rho$ and does not depend on the chosen basis.