Explicit form of the first-passage-time density for accelerating subdiffusion

• We find the Laplace transform of the first passage time density for the accelerating distributed order anomalous diffusion in the case of constant bias.
• We provide an explicit form of the first passage time density for the accelerating distributed order anomalous diffusion in the case of linear bias.
• We compare our theoretical results with numerical simulations for the corresponding subordinated stochastic models.

The first passage time (FPT) statistics play a central role in different fields. Apart of description of the underlying system state and its evolution it is essential to investigate the time at which this state reaches a certain area for the first time. In this paper we study the FPT problem for recently developed models of an accelerating subdiffusion which extends, incorporating various short and long time scalings of the mean square displacement (MSD), popular models of subdiffusion. Based on fractional-order differential equations we derive the Laplace transform of the FPT density function in the case of constant bias. For a force depending linearly on the position we provide a full analytical formula for the FPT density in terms of generalized Mittag-Leffler functions and the Hermite polynomials. We also provide a numerical evidence that FPT properties are strictly connected with MSD scalings of the accelerating subdiffusion.