Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 4

The Zero-boundary Gaussian free field on the unit disc is the centered Gaussian process indexed by finite Borel measures of finite Green energy, with covarianceThis covariance determines all its finite-dimensional distributions.

The Domain Markov property of the Gaussian free field says that for every suitable open $U\subset \mathbb{D}$ one can writewhere $h_U$ is a zero-boundary Gaussian free field on $U$, independent of $h^{\mathbb{D}\setminus U}$, while $h^{\mathbb{D}\setminus U}$ is harmonic on $U$ and carries the information from the field outside $U$.

Take $U=B(0,e^{-t})$ in the Domain Markov property of the Gaussian free field. The field $h^{\mathbb{D}\setminus U}$ is a harmonic function throughout $U$. If $s>t$, the circle $\partial B(0,e^{-s})$ lies inside $U$, so the mean value property for harmonic functions gives

Write $h=h_U+h^{\mathbb{D}\setminus U}$ with $U=B(0,e^{-t})$. By part (b), the harmonic part contributes the same value $h^{\mathbb{D}\setminus U}(0)$ to every inner circle average. At radius $e^{-t}$ the zero-boundary part contributes zero; equivalently, take the limit from inner circles and use the assumed continuity. HenceThe field $h_U$ is independent of $h^{\mathbb{D}\setminus U}$, so the increment has the required independence.

The dilation $z\mapsto e^{t}z$ maps $U$ onto the unit disc and maps the circle of radius $e^{-s}$ onto the circle of radius $e^{-(s-t)}$. The Conformal invariance of the two-dimensional Gaussian free field therefore gives

Part (c), iterated over disjoint nested annuli, gives independent increments, and the law $X_s-X_t\stackrel{d}{=}{X}_{s-t}$ gives stationary increments. Every finite vector is jointly Gaussian by the definition of the Zero-boundary Gaussian free field, and a continuous version was assumed. Moreover $X_0=0$, because the field has zero boundary values.

Thus $X$ is a continuous centered Gaussian process with stationary increments and independent increments. Its variance is a continuous additive function on the nonnegative real numbers, so $\mathrm{Var}(X_t)={\sigma }^{2}t$ for some ${\sigma }^2\ge 0$. The Gaussian-process characterization of Brownian motion now givesfor standard Brownian motion $B$. This is the Circle-average process of the Gaussian free field is Brownian motion.