Circle-average process of the Gaussian free field is Brownian motion

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. HenceThe 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.