Continuum Gaussian free field

The continuum Gaussian free field on a planar domain $D$ is a centered Gaussian process indexed by finite-energy measures or test functions, with covariance given by the Green function of the Laplacian on $D$.

If $\varphi :D\to \tilde{D}$ is a conformal map, pulling back a zero-boundary Gaussian free field on $\tilde{D}$ gives a zero-boundary Gaussian free field on $D$. This follows from the conformal invariance of the planar Dirichlet Green function.

For finite Borel measures $\rho ,\nu$ of finite Green energy, the zero-boundary Gaussian free field $h$ on $D$ is characterized bywhere $G_D$ is the zero-Dirichlet Green function of the Laplacian.

For an open $U\subset D$, the field decomposes as $h=h_U+h^{D\setminus U}$, where $h_U$ is an independent zero-boundary Gaussian free field on $U$ and $h^{D\setminus U}$ is harmonic on $U$ and agrees with the outside field in the distributional sense.

Let ${\rho }_t$ be uniform measure on the circle of radius $e^{-t}$ around the origin in the unit disc. The process $X_t=(h,{\rho }_t)$ is the circle-average process. The Domain Markov property of the Gaussian free field and the mean value property for harmonic functions imply that its increments are independent and stationary.

The circle-average process of a zero-boundary Gaussian free field in the unit disc has the law of a constant multiple of Brownian motion. It is a continuous centered Gaussian process with stationary increments and independent increments and starts at zero.