= Circle-average process of the Gaussian free field is Brownian motion
{c}
In the logarithmic normalization of the <Circle-average process of the Gaussian free field>, the circular averages form standard <Brownian motion> in logarithmic radius. Let $r_s=e^{-s}$ and $r_t=e^{-t}$. Averaging $\log|1-z\overline w|$ over the two circles gives zero by the <mean value property for harmonic functions>. The average of $\log|z-w|$ is $\log\max(r_s,r_t)$, including equal radii by integrability of the logarithmic singularity. Hence
$$
\operatorname{Cov}(X_s,X_t)=-\log\max(e^{-s},e^{-t})=\min(s,t).
$$
The pairings are jointly Gaussian and centered. Their increments on disjoint logarithmic-time <intervals> have zero <covariance>, hence are <independent>. Also $\mathbb E|X_t-X_s|^4=3|t-s|^2$, so the <Kolmogorov continuity theorem> supplies a <continuous modification>. Together with the limiting value $X_0=0$, this proves the <Brownian motion> law.
The result describes one nested family of circle averages about a fixed center, not the field on every spatial curve as a single Brownian path. If the Green kernel is multiplied by $c>0$, the <covariance> becomes $c\min(s,t)$ and $X/\sqrt c$ is standard <Brownian motion>.
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