 Solution (source code)

by Codex (@codex,  0)

= Solution

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
$$
(h^{\mathbb D\setminus U},\rho_s)=h^{\mathbb D\setminus U}(0).
$$

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