Occupation time statistics for non-Markovian random walks

TL;DR

This paper investigates the statistical properties of occupation times for non-Markovian random walks, including cases with stochastic resetting, deriving key equations and analyzing ergodic behavior and distribution laws.

Contribution

It introduces a generalized framework for analyzing occupation times in non-Markovian and reset random walks, extending classical results like the arcsine law.

Findings

Derived and solved the backward Feynman-Kac equation for occupation times.

Analyzed the PDFs, moments, and ergodic properties of occupation times.

Confirmed theoretical results with numerical simulations.

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 behaviour of the PDFs, 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 famous arcsine law and examine its validity pertaining to various regimes of the rest period of the walker. Our results have been verified with numerical simulations exhibiting an excellent agreement.