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For the zero-boundary Gaussian free field in the unit disc, choose the logarithmic Green normalizationFor $t>0$, let ${\rho }_t$ be the uniform probability measure on the circle of radius $e^{-t}$ about zero. The Circle-average process of the Gaussian free field is the measure pairing $X_t=(h,{\rho }_t)$. This is meaningful even though the continuum field has no ordinary pointwise values: the circular measure has finite Green energy, because its diagonal logarithmic singularity is integrable.

Decreasing radius means increasing logarithmic time. Angular averaging of the Green kernel gives $\mathbb{E}[X_s{X}_t]=\min (s,t)$. Thus $X_t\to 0$ in L2 space as $t\downarrow 0$, and define $X_0=0$ by this limit; it is not a pointwise field restriction to the unit-disc boundary. The Circle-average process of the Gaussian free field is Brownian motion article verifies the covariance and continuity. For another positive Green normalization the same process is rescaled by its square root.^[ref]

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.