Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 203 / 3

For real test functions $f,g\in C_c^{\infty }(D)$, the Dirichlet inner product in the normalization compatible with the logarithmic Green kernel isIts norm measures gradient energy rather than the values of the functions. A change of variables shows that this energy is invariant under planar conformal maps: the squared derivative in the transformation of the gradient cancels the area Jacobian determinant.

The convention used for planar Gaussian free field theory is the Dirichlet energy space:It is a real Hilbert space with the extended Dirichlet inner product. A conformal map from the unit disc onto $D$ identifies the two energy completions, so local representatives can be understood through the disc's zero-boundary space.

For an arbitrary unbounded domain this is the homogeneous energy completion; it should be distinguished from completion in the inhomogeneous norm $\Vert f{\Vert }_{L^2}+\Vert \nabla f{\Vert }_{L^2}$ used in another standard definition of the zero-boundary Sobolev space. On bounded domains with a Poincaré inequality, the two norms are equivalent.

Choose an orthonormal basis $(e_n)$ of the Dirichlet energy space and independent variables $({\xi }_n)$ with the standard normal distribution. The zero-boundary Gaussian free field is the formal seriesIts precise energy-indexed interpretation is the isonormal Gaussian processFor every fixed $f$, this sum converges in $L^2$ of the probability space by Parseval identity, and it is a centered Gaussian random variable. For $f,g\in H_0^1(D)$,This covariance characterizes its law independently of the chosen basis. The series does not define an $H_0^1$-valued random element: ${\sum }_n{\xi }_n^2=\infty$ almost surely. It is realized as a random distribution, and its pairing with ordinary test functions is constructed in part (b).

The expression is a Test-function pairing with a Gaussian free field, rather than the pointwise integral of an ordinary random function. For a real test function $\varphi$, take the Dirichlet solutionusing the Green-kernel identity allowed in the original PDF. In particular, $-\Delta f_{\varphi }=2\pi \varphi$. Integration by parts gives, initially for smooth compactly supported $u$ and then by continuity in the Dirichlet energy space,The integral functional is continuous in the energy norm: pull back to the unit disc, where the transformed test function has compact support, and use the Poincaré inequality. Hence $f_{\varphi }$ is its energy-space representer. DefineBy the isonormal Gaussian process definition, it has mean zero and varianceThe logarithmic singularity is locally integrable, so this variance is finite for smooth compactly supported test functions. ThusThe local TeX corrupted the double integral and the bracketed Green-kernel identity; the PDF supplies the identity above and does not assert an infinite variance.

Set $F(w)=(\varphi (w)-\varphi (0))/w$ for $w\ne 0$. The singularity is removable, with $F(0)={\varphi }^{\prime }(0)$. Injectivity of the conformal map makes $F(w)\ne 0$ away from zero, and conformality makes $F(0)\ne 0$ as well. Since the disc is simply connected, $F$ has a holomorphic logarithm. Therefore $\psi =\log |F|$ is a harmonic function, including at zero, and $\psi (0)=\log |{\varphi }^{\prime }(0)|$.

The mean value property for harmonic functions gives, for every $0<r<1$,For general $D$, boundary values mean radial limits, rather than a continuous extension of $\varphi$ to every point of the circle. The boundary logarithmic mean of a univalent function justifies taking $r\uparrow 1$: the Koebe distortion theorem bounds $|F|$ below by $|{\varphi }^{\prime }(0)|/4$, while the standard integral-mean bound for a univalent function, ${\sup }_{r<1}\int |\varphi (r{e}^{i\theta }){|}^{p}d\theta <\infty$ for $0<p<1/2$, gives uniform integrability of its positive logarithm. Radial limits exist almost everywhere, and passage to the integral follows. Thus, writing $dw=d\theta$ for arc length on the unit circle,The expression on the left is the logarithm of the conformal radius of $D$ at $z=\varphi (0)$.

The canonical Dirichlet Green function on the unit disc with its pole at zero is $-\log |w|$. Conformal invariance of the planar Green kernel therefore givesComparing with $G_D(z,y)=-\log |z-y|-G_z(y)$ yieldsThis also explains the boundary data: when $|w|=1$, the correction is $-\log |\varphi (w)-z|$. The composition with ${\varphi }^{-1}$ is harmonic by conformal invariance of harmonicity. Its apparent singularity at $z$ is removable because the quotient defining $\psi$ extends to ${\varphi }^{\prime }(0)$.

Evaluating there proves the regular part of the planar Dirichlet Green function formulaOn an unbounded domain, boundary values alone need not specify a unique harmonic function. Here $G_z$ is the correction belonging to the canonical Dirichlet Green kernel, which fixes that ambiguity.