Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 5 / b

Reflection in $L$ interchanges $x$ and $y$. Relabel the two points if necessary so that $x$ lies on the side of $L$ containing the center $a$; the absolute difference is symmetric in $x$ and $y$. Couple a Brownian motion $B$ started at $x$ to one $B^{\prime }$ started at $y$ by setting $B_t^{\prime }=\varphi (B_t)$ before $T_L$ and $B_t^{\prime }=B_t$ afterwards. The Brownian reflection coupling has the correct marginal laws because reflection is an isometry and the Strong Markov property applies at $T_L$. Once the paths meet on $L$, they agree forever.

If the path from $x$ does not hit the target circle before $T_L$, the coupled paths meet before the relevant uncoupled hitting outcomes can differ. The coupling inequality for total variation therefore gives