  We construct radial stochastic Loewner evolution in multiply connected
domains, choosing the unit disk with concentric circular slits as a family of
standard domains. The natural driving function or input is a diffusion on the
associated Teichm\"uller space. The diffusion stops when it reaches the
boundary of the Teichm\"uller space. We show that for this driving function the
family of random growing compacts has a phase transition for $\kappa=4$ and
$\kappa=8$, and that it satisfies locality for $\kappa=6$.
