Observation time dependent mean first passage time of diffusion and sub-diffusion processes

TL;DR

This paper examines how the mean first passage time (MFPT) for diffusion and subdiffusion processes depends on finite observation time T, revealing linear T dependence at small T and model-sensitive behavior at large T, with implications for experimental analysis.

Contribution

It provides a comprehensive analysis of the observation time dependence of MFPT across multiple diffusion models, highlighting its potential as a sensitive measure of stochastic properties.

Findings

MFPT is linearly dependent on T at small T for all models

Large T behavior of MFPT varies with the stochastic model

Observation time dependence is more sensitive than mean square displacement

Abstract

The mean first passage time, one of the important characteristics for a stochastic process, is often calculated assuming the observation time is infinite. However, in practice, the observation time, T, is always finite and the mean first passage time (MFPT) is dependent on the length of the observation time. In this work, we investigate the observation time dependence of the MFPT of a particle freely moving in the interval [-L,L] for a simple diffusion model and four different models of subdiffusion, the fractional diffusion equation (FDE), scaled Brown motion (SBM), fractional Brownian motion (FBM), and stationary Markovian approximation model of SBM and FBM. We find that the MFPT is linearly dependent on T in the small T limit for all the models investigated, while the large-T behavior of the MFPT is sensitive to stochastic properties of the transport model in question. We also discuss the relationship between the observation time, T, dependence and the travel-length, L, dependence of the MFPT. Our results suggest the observation time dependency of the MFPT can serve as an experimental measure that is far more sensitive to stochastic properties of transport processes than the mean square displacement.