Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 2 / a / Solution

Let $B$ be planar Brownian motion started at $z\in D$, let ${\tau }_D$ be its exit time, and let $\varphi :D\to D^{\prime }$ be conformal. DefineThe conformal invariance of planar Brownian motion states thatis Brownian motion started at $\varphi (z)$ and stopped when it exits $D^{\prime }$. Thus conformal maps preserve Brownian paths after this random time-change.