= Solution
Part (c), iterated over disjoint nested annuli, gives <independent increments>, and the law $X_s-X_t\overset d=X_{s-t}$ 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 $\operatorname{Var}(X_t)=\sigma^2t$ for some $\sigma^2\geq0$. The <Gaussian-process characterization of Brownian motion> now gives
$$
(X_t)_{t\geq0}\overset d=(\sigma B_t)_{t\geq0}
$$
for standard <Brownian motion> $B$. This is the <Circle-average process of the Gaussian free field is Brownian motion>.
Back to article page
