Hitting probability for anomalous diffusion processes

TL;DR

This paper investigates the universal features of hitting probabilities in anomalous diffusion processes, revealing a scaling relation involving the Hurst and persistence exponents, supported by exact calculations and numerical evidence.

Contribution

It establishes a universal scaling law for hitting probabilities in self-affine processes, linking the probability to the Hurst and persistence exponents, and verifies this through exact and numerical methods.

Findings

Hitting probability scales as a power law near zero with exponent =/

The scaling function depends only on the ratio x/L

Results are confirmed in processes including diffusion in disordered potentials

Abstract

We present the universal features of the hitting probability $Q(x,L)$, the probability that a generic stochastic process starting at $x$ and evolving in a box $[0,L]$ hits the upper boundary $L$ before hitting the lower boundary at 0. For a generic self-affine process (describing, for instance, the polymer translocation through a nanopore) we show that $Q(x,L)=Q(x/L)$ and the scaling function $Q(z)\sim z^\phi$ as $z\to 0$ with $\phi=\theta/H$ where $H$ and $\theta$ are respectively the Hurst exponent and the persistence exponent of the process. This result is verified in several exact calculations including when the process represents the position of a particle diffusing in a disordered potential. We also provide numerical supports for our analytical results.