Conformal invariance of planar Brownian motion

If $B$ is planar Brownian motion in a domain $D$ and $f:D\to D^{\prime }$ is a nonconstant holomorphic function, then $f(B)$, run with the clock ${\int }_0^t|f^{\prime }(B_s){|}^{2}ds$, is planar Brownian motion in $D^{\prime }$ until the relevant exit time. In particular, conformal maps transport harmonic measure between planar domains.