Occupation-time statistics for non-Markovian random walks.

Abstract

We study the occupation time statistics for non-Markovian random walkers based on the formalism of the generalized master equation for the continuous-time random walk. We also explore the case when the random walker additionally undergoes a stochastic resetting dynamics. We derive and solve the backward Feynman-Kac equation to find the characteristic function for the occupation time in an interval and for the half occupation time in the semi-infinite domain. We analyze the behavior of the probability density functions, the moments, the limiting distributions, and the ergodic properties for both occupation times when the underlying random walk is normal or anomalous. For the half occupation time, we revisit the well-known arcsine law and examine its validity pertaining to various regimes of the rest period of the walker. Our results have been verified with Monte Carlo-based numerical simulations exhibiting an excellent agreement.