The <Zero-boundary Gaussian free field> on the unit disc is the centered <Gaussian process> indexed by finite <Borel measures> of finite <Green energy>, with <covariance>
$$
\mathbb E[(h,\rho)(h,\nu)]=\iint_{\mathbb D\times\mathbb D}G_{\mathbb D}(x,y)\rho(dx)\nu(dy).
$$
This covariance determines all its <finite-dimensional distributions>.

The <Domain Markov property of the Gaussian free field> says that for every suitable open $U\subset\mathbb D$ one can write
$$
h=h_U+h^{\mathbb D\setminus U},
$$
where $h_U$ is a zero-boundary Gaussian free field on $U$, independent of $h^{\mathbb D\setminus U}$, while $h^{\mathbb D\setminus U}$ is harmonic on $U$ and carries the information from the field outside $U$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 4 / a / Solution

{wiki}

For times $t_1,\ldots,t_n$, the corresponding finite-dimensional distribution of a <stochastic process> $X$ is the <joint probability distribution> of the <random vector> $(X_{t_1},\ldots,X_{t_n})$. The collection of these distributions records every finite set of coordinates of the process.

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 Incoming links: Finite-dimensional distribution