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

The space of test functions $C_0^{\infty }(D)$ consists of infinitely differentiable real-valued functions whose support of a function is a compact subset of $D$.

Equip $C_0^{\infty }(D)$ with the Dirichlet inner productThe zero-boundary Sobolev space $H_0^1(D)$ is its Hilbert space completion in the norm $\Vert f{\Vert }_{\nabla }=(f,f{)}_{\nabla }^{1/2}$.

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 satisfyEquivalently, for an orthonormal basis $(e_n)$ of $H_0^1(D)$ and independent standard normal variables $({\xi }_n)$,as a random generalized function.