= Conformal invariance of Brownian loop soup
Let $f:D\to D'$ be a <conformal map> that is a <bijection> between planar <domains>. Mapping every loop of a <Brownian loop soup> in $D$ by $f$ gives a <Brownian loop soup> of the same intensity in $D'$, after changing the clock along each loop. For a Brownian parametrization $\ell(t)$, the new clock is
$$
s(t)=\int_0^t|f'(\ell(r))|^2\,dr.
$$
The statement concerns unrooted loops: choosing a different starting time on the same loop does not create another object. It follows from conformal invariance of <Brownian loop measure> and the image property of a <Poisson point process>.<The Brownian loop soup by Gregory F. Lawler and Wendelin Werner>[Sections 4.1 and 5.3]{ref}
For the dilation $f(z)=az$, spatial lengths multiply by $|a|$ and loop durations by $|a|^2$, while the soup intensity parameter stays fixed. Thus the random geometry has no preferred length scale. Keeping the clock change explicit prevents confusing conformal invariance of the loop ensemble with preservation of each loop's original duration.
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