Finite-Green-energy measure pairing with a Gaussian free field
= Finite-Green-energy measure pairing with a Gaussian free field
{c}
If a measure $\rho$ has finite <Green energy>, then
$$
(h,\rho)=\sum_{n\geq1}\xi_n\int_De_n\,d\rho
$$
converges 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.