  We propose a model for anomalous transport in inhomogeneous environments,
such as fractured rocks, in which particles move only along pre-existing
self-similar curves (cracks). The stochastic Loewner equation is used to
efficiently generate such curves with tunable fractal dimension $d_f$. We
numerically compute the probability of first passage (in length or time) from
one point on the edge of the semi-infinite plane to any point on the
semi-circle of radius $R$. The scaled probability distributions have a variance
which increases with $d_f$, a non-monotonic skewness, and tails that decay
faster than a simple exponential. The latter is in sharp contrast to
predictions based on fractional dynamics and provides an experimental signature
for our model.
