The zero-boundary <Gaussian free field> $h$ on $D$ is the <isonormal Gaussian process> over $H_0^1(D)$: for $f,g\in H_0^1(D)$, the variables $(h,f)_\nabla$ are centered jointly Gaussian and satisfy
$$
\mathbb E[(h,f)_\nabla(h,g)_\nabla]=(f,g)_\nabla.
$$
Equivalently, for an <orthonormal basis> $(e_n)$ of $H_0^1(D)$ and independent standard normal variables $(\xi_n)$,
$$
h=\sum_{n\geq1}\xi_ne_n
$$
as a random <distribution>[generalized function].

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 4 / a / iii / Solution

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An isonormal Gaussian process over a real <Hilbert space> $H$ is a centered <Gaussian process> $(W(h))_{h\in H}$ satisfying
$$
\mathbb E[W(f)W(g)]=\langle f,g\rangle_H.
$$
It turns the geometry of $H$ into the <covariance> structure of a Gaussian family.

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 Incoming links: Isonormal Gaussian process