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

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.