First-passage times of non-Markovian processes: The case of a reflecting boundary

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Abstract
Mean first-passage times (MFPT) of non-Markovian processes driven by Markovian two-state noise of finite correlation time are considered. Absorbing as well as reflecting boundary conditions are constructed, and new results for the first-passage-time density and the MFPT are derived. We extend our study to dichotomic Fokker-Planck processes, i.e., a stochastic dynamics in which the random walker jumps between two different Fokker-Planck processes with a dichotomic noise dynamics. In this general case, too, we derive the boundary conditions explicitly and obtain novel expressions for the MFPT. A number of special cases and limits are considered which elucidate the physics of the more general results. Finally, we consider the problem of bistability driven by dichotomic noise and express the MFPT in terms of the stationary probability density. For the escape rate at weak noise we establish the connection between the MFPT approach and the current overpopulation method.