First passage time distribution of stationary Markovian processes

Abstract

We investigate how the correlation properties of a stationary Markovian stochastic process affect its First Passage Time Distribution (FPTD). With explicit examples, in this paper we show that the tail of the first passage time distribution crucially depends on the correlation properties of the process and it is independent of its stationary distribution. When the process includes an infinite set of time-scales bounded from above, the FPTD shows tails modulated by some exponential decay. In the case when the process is power-law correlated the FPTD shows power-law tails 1/t^ν and therefore the moments ⟨t^n⟩ of the FPTD are finite only when n < ν − 1. The existence of an infinite and unbounded set of time-scales is a necessary but not sufficient condition in order to observe power-law tails in the FPTD. Finally, we give a general result connecting the FPTD of an additive stochastic processes x(t) to the FPTD of a generic process y(t) related by a coordinate transformation y = f (x) to the first one.