First passage time processes and subordinated SLE

We study the first passage time processes of the anomalous diffusion on the self similar curves in two dimensions. The scaling properties of the mean square displacement and mean first passage time of the fractional Brownian motion and subordinated walk on the 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.