Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 5 / A1 / Solution

The continuum Gaussian free field $\Gamma$ on ${\mathbb{R}}^3$ is the centered Gaussian random distribution indexed by smooth compactly supported test functions, with covariancewhere $G$ is the Green kernel of $-\Delta$. Equivalently, its covariance operator is $(-\Delta {)}^{-1}$.