Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 5 / B4 / Solution

Restrict ${\mu }_x$ to outer boundaries that stay in $D$ and surround $y$. By part (B1), this is the image under outer-boundary formation of rooted Brownian loops through $x$ whose outer boundary stays in $D$ and surrounds $y$. But this restriction is exactly the restriction of ${\rho }_D$ to loops surrounding both $x$ and $y$. Interchanging $x$ and $y$ gives the same intrinsic restriction of ${\rho }_D$, so it can equally be represented by outer boundaries of Brownian loops rooted at $y$ that surround $x$.