First passage time processes and subordinated SLE

TL;DR

This paper investigates the first passage time processes of anomalous diffusion on self-similar fractal curves, deriving scaling properties for various stochastic models and introducing a new parametrized SLE framework.

Contribution

It introduces the natural parametrized subordinated SLE (NS-SLE) as a novel mathematical model for diffusion on fractal curves and derives its scaling properties.

Findings

Scaling laws for mean square displacement and first passage time are established for multiple models.

Numerical analysis confirms the scaling behavior of NS-SLE.

Different fractal curves exhibit distinct diffusion characteristics.

Abstract

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.