  We study the first passage time processes of anomalous diffusion on self
similar curves in two dimensions. The scaling properties of the mean square
displacement and mean first passage time of the ballistic motion, fractional
Brownian motion and subordinated walk on different fractal curves (loop erased
random walk, harmonic explorer and percolation front) are derived. We also
define natural parametrized subordinated Schramm Loewner evolution (NS-SLE) as
a mathematical tool that can model diffusion on fractal curves. The scaling
properties of the mean square displacement and mean first passage time for
NS-SLE are obtained by numerical means.
