to ${\mathbb{R}}^d \setminus D$, then the conditional law of $h$ restricted to $D$ is given by a conditionally deterministic function (the harmonic extension of the field from $\partial D$ to $D$)[^5] plus an independent zero-boundary GFF defined on $D$.

In Section \[sec:projection\] we will establish an analogous property that applies when $h$ is an $\operatorname{FGF}_s({\mathbb{R}}^d)$ with $s \geq 0$. Namely, if we are given the restriction of $h$ to ${\mathbb{R}}^d \setminus D$, then the conditional law of $h$ restricted to $D$ is given by a conditionally deterministic function (the so-called [**$s$-harmonic extension**]{} of the field from ${\mathbb{R}}^d \setminus D$ to $D$) plus a random function (the so-called [**zero-boundary-condition $\operatorname{FGF}_s$ on $D$**]{}). If $s \in {\mathbb{N}}$, then the conditionally deterministic function depends on the restriction to $\partial D$ of $h$ and its derivatives up to a certain order. This follows from the fact that $(-\Delta)^s$ is a local operator when $s \in
{\mathbb{N}}$.

As previously mentioned, another generalization of Brownian motion is the fractional Brownian motion (FBM). Fractional Brownian motion appears to have been first introduced by Kolmogorov in 1940 [@kolmogorov1940wienersche], and the term “fractional Brownian motion” was introduced by Mandelbrot and Van Ness in 1968 [@mandelbrot1968fractional]. As motivation, Mandelbrot and 