Zero-boundary Gaussian free field

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 a Gaussian free field with boundary heights $-\lambda$ and $+\lambda$ on the two sides of the starting point, its distinguished zero-height interface has chordal ${\mathrm{SLE}}_4$ law. Conditional on an initial curve segment, the field is a zero-boundary field in the slit domain plus $m_t(z)=\lambda -(2\lambda /\pi )\arg (g_t(z)-W_t)$. Its mean bracket equals minus the variation of its conditional Green covariance. Subtracting the initial harmonic function gives the corresponding coupling to a zero-boundary field.

If a continuous mean martingale $M$ and a nonnegative finite-variation process $V$ satisfy $d⟨M⟩=-dV$, then the Itô formula shows $\exp (iM-V/2)$ is a bounded complex martingale. Its terminal expectation establishes an entire Gaussian distribution with the initial mean and variance. This is stronger than matching only the first two moments in a random-domain field construction.

A continuum field level line is specified by the harmonic boundary heights it creates and the conditional zero-boundary field remaining on each side. Since a Continuum Gaussian free field is a random distribution, this is not the pointwise set of its zeros. For the height jump appropriate to the Green-function normalization, the zero-height line between opposite Dirichlet heights is chordal ${\mathrm{SLE}}_4$.

For a test function $\varphi$ on $D$, let $f_{\varphi }=-2\pi {\Delta }^{-1}\varphi \in H_0^1(D)$. The pairingis a centered Gaussian random variable with variance

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.

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.

A random set is local for a Gaussian free field if, conditionally on the set and its associated harmonic function, the remaining field is an independent zero-boundary field in its complement. This extends the deterministic Domain Markov property of the Gaussian free field to appropriate random revealed domains. Arbitrary field-dependent random sets need not have this property.

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.