= Solution
{c}
<Conformal invariance of planar Brownian motion> means invariance after changing the clock, not preservation of the original speed. If $f:D\to D'$ is a conformal bijection and $Z$ is planar <Brownian motion> stopped on exiting $D$, put
$$
A_t=\int_0^{t\wedge T_D}|f'(Z_s)|^2\,ds.
$$
Then $f(Z_t)$ has the law of planar Brownian motion started at $f(Z_0)$ and run at time $A_t$, up to the corresponding lifetime. Equivalently, composing with the inverse clock produces ordinary Brownian motion in $D'$.
Indeed, the real and imaginary parts of a holomorphic function are harmonic, so <Itô's formula> gives zero drift. The <Cauchy-Riemann equations> give equal coordinate brackets $A_t$ and zero cross bracket. The <Lévy characterization of multidimensional Brownian motion> after time change identifies the image. In particular the <harmonic measure> of boundary arcs is carried to their conformal images.
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