Take $U=B(0,e^{-t})$ in the <Domain Markov property of the Gaussian free field>. The field $h^{\mathbb D\setminus U}$ is a <harmonic function> throughout $U$. If $s>t$, the circle $\partial B(0,e^{-s})$ lies inside $U$, so the <mean value property for harmonic functions> gives
$$
(h^{\mathbb D\setminus U},\rho_s)=h^{\mathbb D\setminus U}(0).
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-203/4/b/solution