Residence time densities for non-Markovian systems. (I). The two-state system

We study dynamical system which makes transitions between two states at random times. We analyze properties of the cumulative time τ spent by the system in a given state up to time T. When the probability density for the residence time in a single sojourn in the given state differs from a negative exponential the system will be non-Markovian. Simple analytical expressions are derived for the Laplace transform with respect to T of moments of the cumulative residence time. An exact Fourier–Laplace transform of the probability densities for τ at a fixed T are also found. It can be inferred from this expression, that at sufficiently large T the probability densities tend towards a Gaussian. The parameters that define the Gaussian are also given.